The requested file was not found.
Please click here to go to the home page.
The requested file was not found.
Please click here to go to the home page.
6 February 2021 | R. Rosa
This is an introductory post. I used to have a blog about homebrewing, created in 2006, in a time where not much information was available in Brazil about homebrewing. So I guess my blog turned out to be of some value. Since then, I was happy to see the homebrewing community grow so large in Brazil.
Today, I decided to revive the blog, but open up the focus to include more math and coding (yes, there were math in the homebrewing blog since brewing involves many physicochemical processes amenable to mathematical modeling). And to broaden the audience, I also decided, although with some reluctance, to write some posts in English, including this introductory one.
For a long time, my personal homepages were all in pure html, sometimes with a little javascript. About two years ago, I revamped it in Python/Flask. Last year, though, I realized the power of the Julia language and decided to rewrite my website in Julia/Franklin.
Franklin is a static-site generator written purely in Julia, made by Thibaut Lienart. It is so much easier than Flask and good enough for me since I don't really need the power of Flask as building a dynamic site. In fact, in Flask, I ended up building a static site out of the dynamic one before publishing it.
With Franklin, the only thing that was bothering me was that it seemed to me that I wouldn't be able to have comments on the blogs, since it is a static site. However, I ended up learning that there are a number of tools and apps to embed a comment section in static sites with javascripts or iframe. With that, Franklin turned out to be just perfect for what I need.
Deploying the site is as simple as pushing the changes to a github repo. For the comments, there are paid and free apps, open and closed source. I chose to use utterances, a free, open-source, no-fuss solution, which is pretty easy to set up, and which keeps all the comments in another github repo (one can use the same repo as that for the website, but I chose to use a different one, for more independence). One drawback is that to post a comment, one needs to have a github account. But I hope that will turn out not to be of too much trouble.
Anyways, that is it for the Introduction.
Cheers!
6 February 2021 | R. Rosa
This is an introductory post. I used to have a blog about homebrewing, created in 2006, in a time where not much information was available in Brazil about homebrewing. So I guess my blog turned out to be of some value. Since then, I was happy to see the homebrewing community grow so large in Brazil.
Today, I decided to revive the blog, but open up the focus to include more math and coding (yes, there were math in the homebrewing blog since brewing involves many physicochemical processes amenable to mathematical modeling). And to broaden the audience, I also decided, although with some reluctance, to write some posts in English, including this introductory one.
For a long time, my personal homepages were all in pure html, sometimes with a little javascript. About two years ago, I revamped it in Python/Flask. Last year, though, I realized the power of the Julia language and decided to rewrite my website in Julia/Franklin.
Franklin is a static-site generator written purely in Julia, made by Thibaut Lienart. It is so much easier than Flask and good enough for me since I don't really need the power of Flask as building a dynamic site. In fact, in Flask, I ended up building a static site out of the dynamic one before publishing it.
With Franklin, the only thing that was bothering me was that it seemed to me that I wouldn't be able to have comments on the blogs, since it is a static site. However, I ended up learning that there are a number of tools and apps to embed a comment section in static sites with javascripts or iframe. With that, Franklin turned out to be just perfect for what I need.
Deploying the site is as simple as pushing the changes to a github repo. For the comments, there are paid and free apps, open and closed source. I chose to use utterances, a free, open-source, no-fuss solution, which is pretty easy to set up, and which keeps all the comments in another github repo (one can use the same repo as that for the website, but I chose to use a different one, for more independence). One drawback is that to post a comment, one needs to have a github account. But I hope that will turn out not to be of too much trouble.
Anyways, that is it for the Introduction.
Cheers!
9 February 2021 | R. Rosa
In many real-world problems, one is interested in estimating certain key quantities related to the problem. For instance, in fluid flows, quantities of interest involve kinetic energy, enstrophy, drag coefficient, energy dissipation rate, and so on. In other applications, one might be interested in mechanical stress, chemical concentration, infected population, pharmaceutical dosage, etc.
Many such problems can be resonably modeled by differential equations, which may, however, exibit complicate, perhaps chaotic dynamics. In those situations, the instantaneous value of certain quantities vary unpredictably in time, but very often their mean value is reasonably steady.
This mean value can be considered in different ways, e.g. as time average, as ensemble average, or as spatial average, and are thus more ameanable to analysis. This article considers ways to estimate time and ensemble averages of certain quantities.
If a model for the problem is available in the form of an ordinary differential equation
where is some locally Lipschitz function acting in some finite-dimensional space , , then, for each , there exists a unique solution with . If we assume all solutions are defined globally in the futures, we obtain a continuous semigroup acting on , with given by .
Given a function representing some "real" quantity we want to measure, the asymptotic superior limit of the time-average of is given (and denoted) by
The idea here is that we would like to find an upper bound for , for all possible initial conditions , in some subset of interest .
Assuming that is continuous, that is positively invariant (i.e. implies , .), and is compact, then is bounded on , and the superior limit above is uniformily bounded in .
The problem, then, is to find the best possible bound C for .
One way to bound is to do what is called energy-type estimates, which amounts to multiplying the equation by appropriate terms aiming to obtain inequalities that eventually lead to an estimate for . Or by variational estimates, introducing an auxiliary function, within a special class of functions, and performing some minimization with respect to the auxiliary function.
Numerically, we can use a Monte-Carlo method and simulate the evolution of the equation in a computer, with randomly chosen initial conditions, and look for the time average over sufficiently long time intervals (sufficiently in the sense, for instance, that the finite-time average does not change much – according to a given error – by increasing the averaging time). Or one can vary the auxiliary function in the special class of functions and look for the best estimate.
More recently, however, a novel method is being used, which is a sort of variational technique, but in a different perspective and leading to an efficient numerical approach. There are some aspects I would like to discuss concerning this method:
It can be seen as a convex minimization problem;
When and are polynomials, the minimization problem can be relaxed to a P-complete problem by looking for a Sum of Squares (SoS) representation of an appropriate term, at the cost of obtaining a larger bound, but which is often near the optimal value;
The original convex minimization problem can be recast into a minimax problem and be showed to indeed yield the optimal estimate;
The optimality result above has been proved first in the finite-dimensional case and, more recently, myself and a co-author extended it to dissipative evolutionary partial differential equations.
Looking for an expression for a nonnegative multivariate real polynomial as a Sum of Squares (SoS) of other polynomials, i.e. for other polynomials , is not a new problem. In 1885, the 21-year old Minkowski made his inaugural dissertation on quadratic polynomials conjecturing that there must exist homogeneous, nonnegative real polynomials of degree in variables, for arbitrary , which are not sums of squares of other homogeneous real polynomials. Hilbert initially attacked Minkowski's claim, but by the end of the presentation Hilbert was convinced that this might indeed be true at least starting with . Three years later, at the age of 26, Hilbert himself proved that the claim is not true for , but that for or for , the set of nonnegative polynomials of degree in variables is indeed strictly larger than the set of sum of squares of polynomials.
Further work on the subject led him to formulate the 17th problem in his list of 23 problems presented in 1900: must every nonnegative homogenous polynomial be expressed as a sum of squares of rational functions?
Hilbert used tools from classical algebraic geometry at that time, without given explicit examples for the problem addressing Minkowski's dissertation. Explicit examples of homogenous polynomials which are not sum of squares of other polynomials were only given in the second half of the 20th century. One famous example is that of Motzkin:
Hilbert's 17th problem was solved in the affimartive by Artin, in 1926. For further historical accounts related to Hilbert's 17th problem, see e.g. Reznick (2000).
More recently, a number of methods to actually test and find whether a given multivariate nonnegative polynomial is a sum of squares of polynomials have been devised (e.g. Shor 1980s, 1990s, Choi, Lam, Reznik 1990s). Then, Parrilo (2000) presented in his PhD thesis, and in subsequent articles (e.g. Parrilo (2003)), several applications to differential equations, including the search for Lyapunov functions and control strategies. By the early 2000s, a number of MATLAB toolbox solvers were already available.
Applications to local stability of PDEs and, in particular to 2D fluid flows were given, respectively by Papachristodoulou and Peet (2006) and Yu, Kashima, Imura (2008).
Finally we get to the articles related to the main motivation for this pots, which is that of globally estimating quantities related to the problem at hand and the global stability of the model.
The first article which seems to exploit the technique of Sum of Squares to the global analysis of PDEs seem to be that of Goulart and Chernyshenko (2012), which considered, in particular, the global stability of fluid flows. This was soon followed by a number of works by various authors: Fantuzzi, Goluskin, Doering, Goulart, Chernyshenko, Huang, Papachristodoulou (2010s), among others (see e.g. Chernyshenko, Goulart, Huang, and Papachristodoulou). These culminated with the work of Tobasco, Goluskin, and Doering (2018) showing that the convex optimization problem can be written as a minimax problem, which can then be proved to yield an optimal result for the estimate of the asymptotic time averages mentioned earlier. In turn, this gives the expectation that relaxing the problem to use the sum of squares approach yields sharp bounds, close to the optimal one.
Now we begin to directly address the points raised above. Let us go back to the setting described earlier and see how the convex optimization problem appears.
One key aspect is to realize that, given any continuously differentiable function , it follows, by the chain rule and integration, that the asymptotic time average of satisfies
Since is assumed to be compact and positively invariant, then is uniformly bounded in , so that
as . Hence, using that a bar denotes limit superior of the time-averages, we write
Since (6) actually holds for the limit itself, not only the superior limit, then we may add it to and have that
on , for arbitrary continuously differentiable function .
Since the above holds for arbitrary such and since the aim is to obtain the best possible , in the sense of being the smallest possible one, we can write the problem of finding such bound for as the minimization problem
However, if we had to check whether the time averages are smaller than or equal to , for every initial condition in , we would actually have much more work than simply checking whether . So, the idea is to require the much stronger condition that , for every point in . Notice this is not a dynamic condition. We are not solving any differential equation. The point is an arbitrary point in . It is a pointwise bound, that certainly implies the time-average bound for any solution starting at the positively invariant set .
It may seem, at first, that, by requiring this stronger condition, we end up with a much worse bound. However, it turns out that the minimization process somehow compensates for that and end up yielding an optimal bound just like we would obtain by requiring only that the time-average be smaller than or equal to . This magic is taken care of by the inclusion of the auxiliary function , which is sometimes called the reservoir function. Notice that the time-average vanishes, but when considering for arbitrary points in , this term, for suitable , can be negative to compensate when is large, and it is allowed to be positive, when is small, such that at the end we find a relatively small bound .
Notice we don't expect to be negative all the time, otherwise would be like a Lyapunov function, or a La Salle-type function, and the solutions would converge to the invariant set included in the set . Some systems do have such a function, but this is not expected to exist in more complicate problems.
Now, by requiring that holds pointwise in all , instead of only along the time average of the trajectories , we arrive at the following minimization problem:
We may rewrite this as
where . This is a convex minimization problem, since the objetive function is linear, and the minimization is sought after within the set , which is convex since is linear and the half plane is convex.
The minimization problem (10) can be NP-hard to compute. However, when the differential equation term and the quantity of interest are polynomials, the minimization problem can be relaxed to a P-complete convex minimization problem by restricting to be a polynomial of a given order, or some special type of polynomial, and requiring that the polynomial be SoS, which certainly implies the condition that it be nonnegative. That might not yield an optimal bound, but it's been show to yield pretty sharp estimates for a number of equations.
This formulation takes the precise form
where above we denote by the set of real polynomials on .
This problem can be regarded as a semidefinite programming. It is similar to linear programming, but in which the first orthant is replace by the cone of positive semidefinite matrices . More precisely, we may start with the primal problem:
where is a given vector, and are given symmetric matrices, and is the decision variable, also assume to be symmetric. The dot product for matrices is element-wise, i.e. , and means that is positive semidefinite, i.e. , for every .
The minization problem above has the dual formulation
where . Any solution of the dual problem is a lower bound for the primal problem, and, conversely, any solution of the primal problem yields an upper bound for the dual problem. In fact, this follows from
Thus,
for any feasible and in each problem.
The question now is how to frame the Sum of Squares problem into a semidefinite programming one. As described in Parrilo (2003), it is possible to write the sum of squares problem in either the primal form or the dual form. In theory, they are mathematically equivalent, but one formulation may be numerically more efficient than the other, depending on the dimension of the problem. For the sake of illustration, we describe below how to arrive at the primal problem.
So, suppose a multivariate real polynomial , , of degree is given. It is easy to argue that, for to have any chance of being a sum of squares, or just nonnegative, the degree of has to be even, say . It is also not difficult to argue that it can be written in the form
for a symmetric matrix , where
is the vector of all monomials in of degree up to . The dimension of the space for is .
For example, consider the polynomial
in , which we know is SoS since it is precisely . Then, with , we can take
Since the elements of are not algebraically independent (e.g. , such is usually not unique. For instance, we can also take
or any convex combinations of the two.
Back to the general case (16), if there exists a symmetric matrix which is positive semidefinite, then it can be diagonalizable with the elements in the diagonal being all non-negative, i.e.
where is a vector of polynomials in . This yields that is a SoS.
Hence, the problem becomes to find a symmetric positive semidefinite matrix satisfying (16). The polynomials and are equal if, and only if, their coefficients are equal, which is a linear problem for , with dimension . If we define the coeficients of by and those of by , , then the problem becomes to find a symmetrix matrix such that
If we further ask to minimize the quantity for some desirable symmetric matrix , then we end up with the primal semidefinite programming problem for .
The convex minimization problem (10) can easily be rewritten in the minimax form
With this formulation in mind, Tobasco, Goluskin and Doering (2018) gave a beautiful proof that the bound is actually optimal, and that the supremum at the left hand side above is achieved!:
The proof uses Ergodic Theory and a minimax principle. In a future opportunity we will go through its proof, as well as to detail the extension done to the infinite-dimensional setting, which is briefly discussed next.
The proof in the finite dimensional case uses a few conditions that are delicate to extend to the infinite dimensional case:
The positively invariant set has to be compact;
The quantity of interest has to be a continuous function on the phase space ;
Borel probability mesure are Lagrangian invariant if and only if they are Eulerian invariant.
By Lagrangian invariant we mean the classical invariant condition , for any Borel set , where is the Borel probability measure in question and is the semigroup generated by the equation. By Eulerian invariant we mean that has to satisfy , for all .
The assumption that be finite-dimensional is not a requirement per se, but it makes the above conditions hold in more generality. For instance, it suffices to have closed and bounded to have it compact. And this compactness is needed both for the passage from time average to ensemble average (i.e. average with respect to the invariant measure) and for the minimax principle.
Concerning the assumption of continuity of , it is not a big deal in finite dimensions, but it is quite restrictive for partial differential equations. For instance, if the phase space is , one cannot consider involving derivatives of . Even if we attempt to use extensions of the minimax principle, they require upper-semicontinuity of , so even quantities like would not work as is.
But at least for the case of a continuous quantity in the infinite-dimensional (e.g. kinetic energy on ), one can go around the requirement of being compact by considering dissipative systems which possess a compact attracting set.
The remaining delicate condition is the equivalence between Lagrangian and Eulerian invariance, which is by no means trivial in the infinite-dimensional case. In fact, I know of only two equations for which this has been proved: the two-dimensional Navier-Stokes equations and a globally modified Navier-Stokes equations obtained by truncating the nonlinear term. However, it is my belief that the key tool is simply that it be possible to approximate the system (any solution) by a right-invertible semigroup (e.g. Galerkin approximation or a hyperbolic/wave-type approximation) and exploit the usual a~priori estimates. It is an open field to prove this for other systems or to come up with an easily-applicable general statement.
It should be said that even the notion of Eulerian invariance needs to be relaxed to working for special types of functions , which we call cylindrical test functionals. They are at the core of the notion of statistical solution.
As in the finite-dimensional case, we leave further details about the result in infinite dimensions to a future post. Meanwhile, the details can be found in Rosa and Temam (arxiv 2020)
Selected References:
9 February 2021 | R. Rosa
In many real-world problems, one is interested in estimating certain key quantities related to the problem. For instance, in fluid flows, quantities of interest involve kinetic energy, enstrophy, drag coefficient, energy dissipation rate, and so on. In other applications, one might be interested in mechanical stress, chemical concentration, infected population, pharmaceutical dosage, etc.
Many such problems can be resonably modeled by differential equations, which may, however, exibit complicate, perhaps chaotic dynamics. In those situations, the instantaneous value of certain quantities vary unpredictably in time, but very often their mean value is reasonably steady.
This mean value can be considered in different ways, e.g. as time average, as ensemble average, or as spatial average, and are thus more ameanable to analysis. This article considers ways to estimate time and ensemble averages of certain quantities.
If a model for the problem is available in the form of an ordinary differential equation
where is some locally Lipschitz function acting in some finite-dimensional space , , then, for each , there exists a unique solution with . If we assume all solutions are defined globally in the futures, we obtain a continuous semigroup acting on , with given by .
Given a function representing some "real" quantity we want to measure, the asymptotic superior limit of the time-average of is given (and denoted) by
The idea here is that we would like to find an upper bound for , for all possible initial conditions , in some subset of interest .
Assuming that is continuous, that is positively invariant (i.e. implies , .), and is compact, then is bounded on , and the superior limit above is uniformily bounded in .
The problem, then, is to find the best possible bound C for .
One way to bound is to do what is called energy-type estimates, which amounts to multiplying the equation by appropriate terms aiming to obtain inequalities that eventually lead to an estimate for . Or by variational estimates, introducing an auxiliary function, within a special class of functions, and performing some minimization with respect to the auxiliary function.
Numerically, we can use a Monte-Carlo method and simulate the evolution of the equation in a computer, with randomly chosen initial conditions, and look for the time average over sufficiently long time intervals (sufficiently in the sense, for instance, that the finite-time average does not change much – according to a given error – by increasing the averaging time). Or one can vary the auxiliary function in the special class of functions and look for the best estimate.
More recently, however, a novel method is being used, which is a sort of variational technique, but in a different perspective and leading to an efficient numerical approach. There are some aspects I would like to discuss concerning this method:
It can be seen as a convex minimization problem;
When and are polynomials, the minimization problem can be relaxed to a P-complete problem by looking for a Sum of Squares (SoS) representation of an appropriate term, at the cost of obtaining a larger bound, but which is often near the optimal value;
The original convex minimization problem can be recast into a minimax problem and be showed to indeed yield the optimal estimate;
The optimality result above has been proved first in the finite-dimensional case and, more recently, myself and a co-author extended it to dissipative evolutionary partial differential equations.
Looking for an expression for a nonnegative multivariate real polynomial as a Sum of Squares (SoS) of other polynomials, i.e. for other polynomials , is not a new problem. In 1885, the 21-year old Minkowski made his inaugural dissertation on quadratic polynomials conjecturing that there must exist homogeneous, nonnegative real polynomials of degree in variables, for arbitrary , which are not sums of squares of other homogeneous real polynomials. Hilbert initially attacked Minkowski's claim, but by the end of the presentation Hilbert was convinced that this might indeed be true at least starting with . Three years later, at the age of 26, Hilbert himself proved that the claim is not true for , but that for or for , the set of nonnegative polynomials of degree in variables is indeed strictly larger than the set of sum of squares of polynomials.
Further work on the subject led him to formulate the 17th problem in his list of 23 problems presented in 1900: must every nonnegative homogenous polynomial be expressed as a sum of squares of rational functions?
Hilbert used tools from classical algebraic geometry at that time, without given explicit examples for the problem addressing Minkowski's dissertation. Explicit examples of homogenous polynomials which are not sum of squares of other polynomials were only given in the second half of the 20th century. One famous example is that of Motzkin:
Hilbert's 17th problem was solved in the affimartive by Artin, in 1926. For further historical accounts related to Hilbert's 17th problem, see e.g. Reznick (2000).
More recently, a number of methods to actually test and find whether a given multivariate nonnegative polynomial is a sum of squares of polynomials have been devised (e.g. Shor 1980s, 1990s, Choi, Lam, Reznik 1990s). Then, Parrilo (2000) presented in his PhD thesis, and in subsequent articles (e.g. Parrilo (2003)), several applications to differential equations, including the search for Lyapunov functions and control strategies. By the early 2000s, a number of MATLAB toolbox solvers were already available.
Applications to local stability of PDEs and, in particular to 2D fluid flows were given, respectively by Papachristodoulou and Peet (2006) and Yu, Kashima, Imura (2008).
Finally we get to the articles related to the main motivation for this pots, which is that of globally estimating quantities related to the problem at hand and the global stability of the model.
The first article which seems to exploit the technique of Sum of Squares to the global analysis of PDEs seem to be that of Goulart and Chernyshenko (2012), which considered, in particular, the global stability of fluid flows. This was soon followed by a number of works by various authors: Fantuzzi, Goluskin, Doering, Goulart, Chernyshenko, Huang, Papachristodoulou (2010s), among others (see e.g. Chernyshenko, Goulart, Huang, and Papachristodoulou). These culminated with the work of Tobasco, Goluskin, and Doering (2018) showing that the convex optimization problem can be written as a minimax problem, which can then be proved to yield an optimal result for the estimate of the asymptotic time averages mentioned earlier. In turn, this gives the expectation that relaxing the problem to use the sum of squares approach yields sharp bounds, close to the optimal one.
Now we begin to directly address the points raised above. Let us go back to the setting described earlier and see how the convex optimization problem appears.
One key aspect is to realize that, given any continuously differentiable function , it follows, by the chain rule and integration, that the asymptotic time average of satisfies
Since is assumed to be compact and positively invariant, then is uniformly bounded in , so that
as . Hence, using that a bar denotes limit superior of the time-averages, we write
Since (6) actually holds for the limit itself, not only the superior limit, then we may add it to and have that
on , for arbitrary continuously differentiable function .
Since the above holds for arbitrary such and since the aim is to obtain the best possible , in the sense of being the smallest possible one, we can write the problem of finding such bound for as the minimization problem
However, if we had to check whether the time averages are smaller than or equal to , for every initial condition in , we would actually have much more work than simply checking whether . So, the idea is to require the much stronger condition that , for every point in . Notice this is not a dynamic condition. We are not solving any differential equation. The point is an arbitrary point in . It is a pointwise bound, that certainly implies the time-average bound for any solution starting at the positively invariant set .
It may seem, at first, that, by requiring this stronger condition, we end up with a much worse bound. However, it turns out that the minimization process somehow compensates for that and end up yielding an optimal bound just like we would obtain by requiring only that the time-average be smaller than or equal to . This magic is taken care of by the inclusion of the auxiliary function , which is sometimes called the reservoir function. Notice that the time-average vanishes, but when considering for arbitrary points in , this term, for suitable , can be negative to compensate when is large, and it is allowed to be positive, when is small, such that at the end we find a relatively small bound .
Notice we don't expect to be negative all the time, otherwise would be like a Lyapunov function, or a La Salle-type function, and the solutions would converge to the invariant set included in the set . Some systems do have such a function, but this is not expected to exist in more complicate problems.
Now, by requiring that holds pointwise in all , instead of only along the time average of the trajectories , we arrive at the following minimization problem:
We may rewrite this as
where . This is a convex minimization problem, since the objetive function is linear, and the minimization is sought after within the set , which is convex since is linear and the half plane is convex.
The minimization problem (10) can be NP-hard to compute. However, when the differential equation term and the quantity of interest are polynomials, the minimization problem can be relaxed to a P-complete convex minimization problem by restricting to be a polynomial of a given order, or some special type of polynomial, and requiring that the polynomial be SoS, which certainly implies the condition that it be nonnegative. That might not yield an optimal bound, but it's been show to yield pretty sharp estimates for a number of equations.
This formulation takes the precise form
where above we denote by the set of real polynomials on .
This problem can be regarded as a semidefinite programming. It is similar to linear programming, but in which the first orthant is replace by the cone of positive semidefinite matrices . More precisely, we may start with the primal problem:
where is a given vector, and are given symmetric matrices, and is the decision variable, also assume to be symmetric. The dot product for matrices is element-wise, i.e. , and means that is positive semidefinite, i.e. , for every .
The minization problem above has the dual formulation
where . Any solution of the dual problem is a lower bound for the primal problem, and, conversely, any solution of the primal problem yields an upper bound for the dual problem. In fact, this follows from
Thus,
for any feasible and in each problem.
The question now is how to frame the Sum of Squares problem into a semidefinite programming one. As described in Parrilo (2003), it is possible to write the sum of squares problem in either the primal form or the dual form. In theory, they are mathematically equivalent, but one formulation may be numerically more efficient than the other, depending on the dimension of the problem. For the sake of illustration, we describe below how to arrive at the primal problem.
So, suppose a multivariate real polynomial , , of degree is given. It is easy to argue that, for to have any chance of being a sum of squares, or just nonnegative, the degree of has to be even, say . It is also not difficult to argue that it can be written in the form
for a symmetric matrix , where
is the vector of all monomials in of degree up to . The dimension of the space for is .
For example, consider the polynomial
in , which we know is SoS since it is precisely . Then, with , we can take
Since the elements of are not algebraically independent (e.g. , such is usually not unique. For instance, we can also take
or any convex combinations of the two.
Back to the general case (16), if there exists a symmetric matrix which is positive semidefinite, then it can be diagonalizable with the elements in the diagonal being all non-negative, i.e.
where is a vector of polynomials in . This yields that is a SoS.
Hence, the problem becomes to find a symmetric positive semidefinite matrix satisfying (16). The polynomials and are equal if, and only if, their coefficients are equal, which is a linear problem for , with dimension . If we define the coeficients of by and those of by , , then the problem becomes to find a symmetrix matrix such that
If we further ask to minimize the quantity for some desirable symmetric matrix , then we end up with the primal semidefinite programming problem for .
The convex minimization problem (10) can easily be rewritten in the minimax form
With this formulation in mind, Tobasco, Goluskin and Doering (2018) gave a beautiful proof that the bound is actually optimal, and that the supremum at the left hand side above is achieved!:
The proof uses Ergodic Theory and a minimax principle. In a future opportunity we will go through its proof, as well as to detail the extension done to the infinite-dimensional setting, which is briefly discussed next.
The proof in the finite dimensional case uses a few conditions that are delicate to extend to the infinite dimensional case:
The positively invariant set has to be compact;
The quantity of interest has to be a continuous function on the phase space ;
Borel probability mesure are Lagrangian invariant if and only if they are Eulerian invariant.
By Lagrangian invariant we mean the classical invariant condition , for any Borel set , where is the Borel probability measure in question and is the semigroup generated by the equation. By Eulerian invariant we mean that has to satisfy , for all .
The assumption that be finite-dimensional is not a requirement per se, but it makes the above conditions hold in more generality. For instance, it suffices to have closed and bounded to have it compact. And this compactness is needed both for the passage from time average to ensemble average (i.e. average with respect to the invariant measure) and for the minimax principle.
Concerning the assumption of continuity of , it is not a big deal in finite dimensions, but it is quite restrictive for partial differential equations. For instance, if the phase space is , one cannot consider involving derivatives of . Even if we attempt to use extensions of the minimax principle, they require upper-semicontinuity of , so even quantities like would not work as is.
But at least for the case of a continuous quantity in the infinite-dimensional (e.g. kinetic energy on ), one can go around the requirement of being compact by considering dissipative systems which possess a compact attracting set.
The remaining delicate condition is the equivalence between Lagrangian and Eulerian invariance, which is by no means trivial in the infinite-dimensional case. In fact, I know of only two equations for which this has been proved: the two-dimensional Navier-Stokes equations and a globally modified Navier-Stokes equations obtained by truncating the nonlinear term. However, it is my belief that the key tool is simply that it be possible to approximate the system (any solution) by a right-invertible semigroup (e.g. Galerkin approximation or a hyperbolic/wave-type approximation) and exploit the usual a~priori estimates. It is an open field to prove this for other systems or to come up with an easily-applicable general statement.
It should be said that even the notion of Eulerian invariance needs to be relaxed to working for special types of functions , which we call cylindrical test functionals. They are at the core of the notion of statistical solution.
As in the finite-dimensional case, we leave further details about the result in infinite dimensions to a future post. Meanwhile, the details can be found in Rosa and Temam (arxiv 2020)
Selected References:
22 February 2021 | R. Rosa
We addressed, in the previous Time average bounds via Sum of Squares post, the problem of estimating the asymptotic limit of time averages of quantities related to the solutions of a differential equation.
Here, the aim is to consider an example, namely the Van der Pol oscillator, and use two numerical methods to obtain those bounds: via time evolution of the system and via a convex semidefinite programming using Sum of Squares (SoS), both discussed in the previous post.
This example is addressed in details in Fantuzzi, Goluskin, Huang, and Chernyshenko (2016). My main motivation is to visualize the auxiliary function that yields the optimal bound via SoS. That is the main reason to choose a two-dimensional system.
That bound depends on the chosen degree for the auxiliary function appearing in the SoS method. Here are the results for specific values of :
BoundsError: attempt to access 0-element Vector{Any} at index [1]We discuss, below, the details leading to these result and how to appreciate the plots above. (Notice you can rotate, pan and zoom the previous and subsequent images!)
The Van der Pol oscillator originated in the study of eletric circuits and appears in several other phenomena, from control, to biology and seismology.
In its simplest, and nondimensional, form, the equation reads
where . This equation has the stationary solution , , and all other solutions converge to a limit cycle that oscillates around the origin. The form of the limit cycle and the convergence rate to the limit cycle change according to the value of the parameter .
Below you will find the evolution of both and its derivative for different values of , and with the initial conditions and . Notice that, for small , the solution converges faster to the limit cycle, which is nearly a sinusoidal wave; while for large , the solution converges slightly slower and the solution develops spikes, as if firing up some signal information (think neurons in biological applications).
Failed to precompile PlotlyJS [f0f68f2c-4968-5e81-91da-67840de0976a] to "/home/runner/.julia/compiled/v1.10/PlotlyJS/jl_gm5S2j".Another common representation of the dynamics of an autonomous system is that of a phase portrait, in which we draw the orbit, or trajectory, in the phase space of the system. In this case, the phase space is , where . We rewrite the second-order differential equation as a system of first order equations
The same trajectories above, in the two extreme values for , are seen below in phase space.
UndefVarError:PlotlyJS not defined The post Time average bounds via Sum of Squares addressed the problem of estimating the time-average
for the solutions , , of a differential equation
where is continuous; is some locally Lipschitz function acting in some finite-dimensional space , ; and assuming the solutions generate a continuous semigroup , where .
We exemplify this here with the Van der Pol system (2) and with the quantity
The most direct way to numerically estimate the bound is via a numerical evolution of the system, for a sufficiently long time, and taking the corresponding time average. Since the system has a globally attracting limit cycle (except for the unstable fixed point at the origin), we may simply consider a single trajectory for the estimate.
UndefVarError:DifferentialEquations not defined Notice we integrated for a very long time to have a proper convergence of the time average. Alternatively, knowning that a fixed fraction of the time integral over the time period converges to zero, we could start integrating at a later time and avoid the initial large values, due to the transient behavior, and the initial bumps, due to the periodic spikes in the solution and its derivative:
In general, however, we may not have such a clear understanding of the dynamics so as to start from specific points or at specific times.
We now estimate the upper bound via optimization. As we have seen in Time average bounds via Sum of Squares, since and are polynomials, an upper bound is given by
where denotes the set of real polynomials on with degree at most , and where, in this case, . We expect the bound to become sharper as we increase the degree . Recall, as discussed in the previous post, the degree has to be even, otherwise it has no chance of being nonnegative.
Below is the result of the estimate, for some choices of :
MethodError: no method matching prettytable(::IOBuffer, ::Matrix{Float64}, ::Matrix{String}; backend::Symbol, standalone::Bool, formatters::Main.FDSANDBOX_2569371999895574908.var"#17#18")Closest candidates are: prettytable(::IO, ::Any; header, kwargs...) @ PrettyTables ~/.julia/packages/PrettyTables/E8rPJ/src/print.jl:740 prettytable(!Matched::Type{HTML}, ::Any; kwargs...) @ PrettyTables ~/.julia/packages/PrettyTables/E8rPJ/src/print.jl:728 pretty_table(!Matched::Type{String}, ::Any; color, kwargs...) @ PrettyTables ~/.julia/packages/PrettyTables/E8rPJ/src/print.jl:722 ...
Here is a corresponding plot, but skipping , for scaling reasons:
UndefVarError:json not defined Looking at (7) and its feasibility condition
we see that is smaller, the greater the term , i.e. the "closer" points to , or, in other words, the faster the orbit descends along , whenever possible.
The best situation is in a gradient flow, but that is not always the case. It may not even be possible to have descend along all the time; just think of a periodic orbit, with a nontrivial auxiliary function , such as in our case. Nevertheless, the longer the orbit descends along , the better.
With that in mind, you may go back to the visualizations of the auxiliary function given in the beginning of the post. Notice how the condition just described (of the orbit to attempt to descend along the optimal ) improves as the degree of is allowed to increase. You can rotate, pan, and zoom the figures as you wish, to better observe this behavior of the limit cycle with respect to .
We show below a simple performance comparison between the time integration method done in the function get_means_vdp() and the SoS optimization method done in get_bound_vdp_sos() (see the codes in Section Julia codes). For that, we use the JuliaCI/BenchmarkTools.jl package. Keep in mind the functions being compared have not been optimized and this is not a thorough assessment of both methods.
For the time-integration, instead of integrating from to , with , as done in one of the plots above, we only go up to , which is enough to get to the same bound as that given by the SoS method with the auxiliary function with degree . Here is the result of @btime.
julia> using BenchmarkTools
+ Computing time average bounds for the Van der Pol oscillator in Julia Computing time average bounds for the Van der Pol oscillator in Julia
22 February 2021 | R. Rosa
Introduction
We addressed, in the previous Time average bounds via Sum of Squares post, the problem of estimating the asymptotic limit of time averages of quantities related to the solutions of a differential equation.
Here, the aim is to consider an example, namely the Van der Pol oscillator, and use two numerical methods to obtain those bounds: via time evolution of the system and via a convex semidefinite programming using Sum of Squares (SoS), both discussed in the previous post.
This example is addressed in details in Fantuzzi, Goluskin, Huang, and Chernyshenko (2016). My main motivation is to visualize the auxiliary function that yields the optimal bound via SoS. That is the main reason to choose a two-dimensional system.
That bound depends on the chosen degree for the auxiliary function appearing in the SoS method. Here are the results for specific values of :
BoundsError: attempt to access 0-element Vector{Any} at index [1] We discuss, below, the details leading to these result and how to appreciate the plots above. (Notice you can rotate, pan and zoom the previous and subsequent images!)
The Van der Pol oscillator
The Van der Pol oscillator originated in the study of eletric circuits and appears in several other phenomena, from control, to biology and seismology.
In its simplest, and nondimensional, form, the equation reads
where . This equation has the stationary solution , , and all other solutions converge to a limit cycle that oscillates around the origin. The form of the limit cycle and the convergence rate to the limit cycle change according to the value of the parameter .
Below you will find the evolution of both and its derivative for different values of , and with the initial conditions and . Notice that, for small , the solution converges faster to the limit cycle, which is nearly a sinusoidal wave; while for large , the solution converges slightly slower and the solution develops spikes, as if firing up some signal information (think neurons in biological applications).
Failed to precompile PlotlyJS [f0f68f2c-4968-5e81-91da-67840de0976a] to "/home/runner/.julia/compiled/v1.10/PlotlyJS/jl_wPrQD1". Another common representation of the dynamics of an autonomous system is that of a phase portrait, in which we draw the orbit, or trajectory, in the phase space of the system. In this case, the phase space is , where . We rewrite the second-order differential equation as a system of first order equations
The same trajectories above, in the two extreme values for , are seen below in phase space.
UndefVarError: PlotlyJS not defined Estimating time averages
The post Time average bounds via Sum of Squares addressed the problem of estimating the time-average
for the solutions , , of a differential equation
where is continuous; is some locally Lipschitz function acting in some finite-dimensional space , ; and assuming the solutions generate a continuous semigroup , where .
We exemplify this here with the Van der Pol system (2) and with the quantity
Bounds via direct computation of the trajectory and its time average
The most direct way to numerically estimate the bound is via a numerical evolution of the system, for a sufficiently long time, and taking the corresponding time average. Since the system has a globally attracting limit cycle (except for the unstable fixed point at the origin), we may simply consider a single trajectory for the estimate.
UndefVarError: DifferentialEquations not defined Notice we integrated for a very long time to have a proper convergence of the time average. Alternatively, knowning that a fixed fraction of the time integral over the time period converges to zero, we could start integrating at a later time and avoid the initial large values, due to the transient behavior, and the initial bumps, due to the periodic spikes in the solution and its derivative:
In general, however, we may not have such a clear understanding of the dynamics so as to start from specific points or at specific times.
Bounds via convex semidefinite programming with Sum of Squares
We now estimate the upper bound via optimization. As we have seen in Time average bounds via Sum of Squares, since and are polynomials, an upper bound is given by
where denotes the set of real polynomials on with degree at most , and where, in this case, . We expect the bound to become sharper as we increase the degree . Recall, as discussed in the previous post, the degree has to be even, otherwise it has no chance of being nonnegative.
Below is the result of the estimate, for some choices of :
MethodError: no method matching prettytable(::IOBuffer, ::Matrix{Float64}, ::Matrix{String}; backend::Symbol, standalone::Bool, formatters::Main.FDSANDBOX_2569371999895574908.var"#17#18") Closest candidates are: prettytable(::IO, ::Any; header, kwargs...) @ PrettyTables ~/.julia/packages/PrettyTables/E8rPJ/src/print.jl:740 prettytable(!Matched::Type{HTML}, ::Any; kwargs...) @ PrettyTables ~/.julia/packages/PrettyTables/E8rPJ/src/print.jl:728 pretty_table(!Matched::Type{String}, ::Any; color, kwargs...) @ PrettyTables ~/.julia/packages/PrettyTables/E8rPJ/src/print.jl:722 ...
Here is a corresponding plot, but skipping , for scaling reasons:
UndefVarError: json not defined Visualizing the auxiliary function
Looking at (7) and its feasibility condition
we see that is smaller, the greater the term , i.e. the "closer" points to , or, in other words, the faster the orbit descends along , whenever possible.
The best situation is in a gradient flow, but that is not always the case. It may not even be possible to have descend along all the time; just think of a periodic orbit, with a nontrivial auxiliary function , such as in our case. Nevertheless, the longer the orbit descends along , the better.
With that in mind, you may go back to the visualizations of the auxiliary function given in the beginning of the post. Notice how the condition just described (of the orbit to attempt to descend along the optimal ) improves as the degree of is allowed to increase. You can rotate, pan, and zoom the figures as you wish, to better observe this behavior of the limit cycle with respect to .
Computation comparison
We show below a simple performance comparison between the time integration method done in the function get_means_vdp() and the SoS optimization method done in get_bound_vdp_sos() (see the codes in Section Julia codes). For that, we use the JuliaCI/BenchmarkTools.jl package. Keep in mind the functions being compared have not been optimized and this is not a thorough assessment of both methods.
For the time-integration, instead of integrating from to , with , as done in one of the plots above, we only go up to , which is enough to get to the same bound as that given by the SoS method with the auxiliary function with degree . Here is the result of @btime.
julia> using BenchmarkTools
julia> @btime get_means_vdp($u0, $μ, $ϕ, 610)[2][end]
768.929 ms (3115182 allocations: 77.89 MiB)
@@ -131,4 +131,4 @@
tracelinevxy = PlotlyJS.scatter(;x=vdp_sol_x, y=vdp_sol_y, z=v.(vdp_sol_x,vdp_sol_y), line_width=6, line_color="green", mode="lines", type="scatter3d", name="lifted orbit")
push!(plt_composite, PlotlyJS.Plot([tracesurf,traceline0,tracelinevxy], Layout(;xaxis_title = "x", yaxis_title = "y", zaxis_title="z=ln(1+V-min(V))", legend_x=0.0, legend_y=1.0, title="Auxiliary function V=V(x,y) with degree m=$(Vdeg_range[j]) and bound $(round(bounds[j],digits=3))")))
end
UndefVarError: `vdp_sol` not defined
-
⬇ Download the full julia code 
Acknowledgements
There are many people to thank for, in getting to this point, but I specifically want to thank Chris Rackauckas, for pointing me to use JuliaMath/QuadGK.jl; Eric Hanson, for helping me in using his package ericphanson/SDPAFamily; and Thibaut Lienart for helping me with many of the features in his package tlienart/Franklin.jl.
References
© Ricardo M. S. Rosa. Last modified: March 18, 2024. Built with Franklin.jl.
\ No newline at end of file
+There are many people to thank for, in getting to this point, but I specifically want to thank Chris Rackauckas, for pointing me to use JuliaMath/QuadGK.jl; Eric Hanson, for helping me in using his package ericphanson/SDPAFamily; and Thibaut Lienart for helping me with many of the features in his package tlienart/Franklin.jl.
6 Nov 2022 | R. Rosa
Random ordinary differential equations are directly related to Stochastic differential equations and are used in a number of models in a range of areas.
As in many complicate differential equations, numerical methods are of fundamental importance in the understanding such models and in using them in applications.
6 Nov 2022 | R. Rosa
Random ordinary differential equations are directly related to Stochastic differential equations and are used in a number of models in a range of areas.
As in many complicate differential equations, numerical methods are of fundamental importance in the understanding such models and in using them in applications.
I am a Full Professor at the Applied Mathematics Department (Matemática Aplicada) in the Institute of Mathematics of the Federal University of Rio de Janeiro (Instituto de Matemática/Universidade Federal do Rio de Janeiro - IM/UFRJ), with a PhD degree in Applied Mathematics from the Indiana University, USA.
My background is on Partial Differential Equations, with emphasis in Infinite Dimensional Dynamical Systems, incompressible Navier-Stokes equations, and statistical solutions connected with turbulence.
— Ricardo M. S. Rosa
I am teaching, for the third time, a course on theoretical and numerical aspects of stochastic and random ordinary differential equations. Lecture notes in Portuguese availabe at rmsrosa.github.io/notas_sde.
I am also teaching the usual second-year undergraduate course on ordinary differential equations, with my lecture notes available in Portuguese at Equações Diferenciais versão ago/2022.
Article on the "BR-EMS 2021 life table for the Brazilian insured population" published in January 2023 at the REBEP journal (Revista Brasileira de Estudos de População), stemmed from a research project conducted by the Laboratório de Matemática Aplicada (LabMA/UFRJ) in partnership with Federação Nacional de Previdência Privada e Vida (FENAPREVI).
Article "Remembrances of Ciprian Ilie Foias" published in Notices of the American Mathematical Society.
Here are the slides of my talk at the VI Workshop on Fluids and PDE, Celebrating the 60th birthdays of Helena J. Nussenzveig Lopes and Milton C. Lopes Filho.
I am a Full Professor at the Applied Mathematics Department (Matemática Aplicada) in the Institute of Mathematics of the Federal University of Rio de Janeiro (Instituto de Matemática/Universidade Federal do Rio de Janeiro - IM/UFRJ), with a PhD degree in Applied Mathematics from the Indiana University, USA.
My background is on Partial Differential Equations, with emphasis in Infinite Dimensional Dynamical Systems, incompressible Navier-Stokes equations, and statistical solutions connected with turbulence.
— Ricardo M. S. Rosa
I am teaching a graduate-level course on Linear Algebra. Lecture notes in Portuguese availabe at Álgebra Linear Avançada - version 4/jan/2024.
I am also teaching a course on generative methods based on stochastic models, with my lecture notes available at rmsrosa.github.io/random_notes/dev/generative/overview/.
Article on the "BR-EMS 2021 life table for the Brazilian insured population" published in January 2023 at the REBEP journal (Revista Brasileira de Estudos de População), stemmed from a research project conducted by the Laboratório de Matemática Aplicada (LabMA/UFRJ) in partnership with Federação Nacional de Previdência Privada e Vida (FENAPREVI).
Article "Remembrances of Ciprian Ilie Foias" published in Notices of the American Mathematical Society.
Here are the slides of my talk at the VI Workshop on Fluids and PDE, Celebrating the 60th birthdays of Helena J. Nussenzveig Lopes and Milton C. Lopes Filho.
Sou Professor Titular do Departamento de Matemática Aplicada do IM-UFRJ, com doutorado em Matemática Aplicada pela Universidade de Indiana, nos EUA.
A minha formação é em equações a derivadas parciais, com ênfase em dinâmica em dimensão infinita, equações de Navier-Stokes de escoamentos incompressíveis e soluções estatísticas em turbulência.
— Ricardo M. S. Rosa
Veja aqui informações sobre essa disciplina, período 2024/1 com espelho na pós-graduação.
Veja aqui informações sobre a disciplina de Equações Diferenciais, período 2024/1.
Article on the "BR-EMS 2021 life table for the Brazilian insured population" published in January 2023 at the REBEP journal (Revista Brasileira de Estudos de População), stemmed from a research project conducted by the Laboratório de Matemática Aplicada (LabMA/UFRJ) in partnership with Federação Nacional de Previdência Privada e Vida (FENAPREVI).
Article "Remembrances of Ciprian Ilie Foias" published in Notices of the American Mathematical Society.
Here are the slides of my talk at the VI Workshop on Fluids and PDE, Celebrating the 60th birthdays of Helena J. Nussenzveig Lopes and Milton C. Lopes Filho.
Sou Professor Titular do Departamento de Matemática Aplicada do IM-UFRJ, com doutorado em Matemática Aplicada pela Universidade de Indiana, nos EUA.
A minha formação é em equações a derivadas parciais, com ênfase em dinâmica em dimensão infinita, equações de Navier-Stokes de escoamentos incompressíveis e soluções estatísticas em turbulência.
— Ricardo M. S. Rosa
Veja aqui informações sobre essa disciplina, período 2024/2.
Veja aqui informações sobre a disciplina de Álgebra Linear do Mestrado, período 2024/2, com espelho na graduação Álgebra Linear Avançada.
Article on the "BR-EMS 2021 life table for the Brazilian insured population" published in January 2023 at the REBEP journal (Revista Brasileira de Estudos de População), stemmed from a research project conducted by the Laboratório de Matemática Aplicada (LabMA/UFRJ) in partnership with Federação Nacional de Previdência Privada e Vida (FENAPREVI).
Article "Remembrances of Ciprian Ilie Foias" published in Notices of the American Mathematical Society.
Here are the slides of my talk at the VI Workshop on Fluids and PDE, Celebrating the 60th birthdays of Helena J. Nussenzveig Lopes and Milton C. Lopes Filho.
2023
2022
2021
Buckingham-Pi Theorem and a julia package May 2, 2021
Computing time average bounds for the Van der Pol oscillator in Julia February 22, 2021
Time average bounds via Sum of Squares February 9, 2021
Greetings February 6, 2021
2023
2022
2021
Buckingham-Pi Theorem and a julia package May 2, 2021
Computing time average bounds for the Van der Pol oscillator in Julia February 22, 2021
Time average bounds via Sum of Squares February 9, 2021
Greetings February 6, 2021
Email: rrosa AT im DOT ufrj DOT br
Webpage UFRJ: http://www.im.ufrj.br/rrosa
Office: C-113B, Bloco C, Centro de Tecnologia (CT), Ilha do Fundão
Postal address:
Instituto de Matemática - UFRJ
Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C
Cidade Universitária - Ilha do Fundão - Caixa Postal 68530
Rio de Janeiro - RJ - 21.941-909 - Brasil
Location:
Email: rrosa AT im DOT ufrj DOT br
Webpage UFRJ: http://www.im.ufrj.br/rrosa or https://rmsrosa.github.io
Office: C-127-07, Centro de Tecnologia (CT), Ilha do Fundão
Postal address:
Instituto de Matemática - UFRJ
Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C
Cidade Universitária - Ilha do Fundão - Caixa Postal 68530
Rio de Janeiro - RJ - 21.941-909 - Brasil
Location:
Email profissional: rrosa AT im DOT ufrj DOT br
Webpage UFRJ: http://www.im.ufrj.br/rrosa
Gabinete: C-113B, Bloco C, Centro de Tecnologia (CT), Ilha do Fundão
Endereço Postal:
Instituto de Matemática - UFRJ
Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C
Cidade Universitária - Ilha do Fundão - Caixa Postal 68530
Rio de Janeiro - RJ - 21.941-909 - Brasil
Localização:
Email profissional: rrosa AT im DOT ufrj DOT br
Webpage UFRJ: http://www.im.ufrj.br/rrosa ou https://rmsrosa.github.io
Gabinete: C-127-07, Bloco C, Centro de Tecnologia (CT), Ilha do Fundão
Endereço Postal:
Instituto de Matemática - UFRJ
Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C
Cidade Universitária - Ilha do Fundão - Caixa Postal 68530
Rio de Janeiro - RJ - 21.941-909 - Brasil
Localização:
Destaque Acadêmico do Instituto de Matemática, CCMN-UFRJ, 2009
Prêmio "Antônio Luís Vianna 2000" para docentes recém-doutores - UFRJ/FUJB, Novembro de 2000
Programa "Antônio Luís Vianna 1998" para docentes recém-doutores - UFRJ/FUJB, Outubro de 1998
John Ewing Book Award 1996 - Indiana University (US$300 em livros da Springer-Verlag)
Bolsa de Produtividade em Pesquisa, nível 1, pelo CNPq, ago/2003 até fev/2022
Bolsa de Produtividade em Pesquisa, nível 2B, pelo CNPq, ago/2001 a jul/2003
Bolsa de Produtividade em Pesquisa, nível 2B, pelo CNPq, ago/1999 a jul/2001
Bolsa de Produtividade em Pesquisa, nível 2C, pelo CNPq, ago/1997 a jul/1999
Bolsa de doutorado pelo CNPq na Indiana University, EUA, set/1992 a dez/1996
Bolsa de mestrado pelo CNPq na UFRJ, set/1989 a fev/1992
Bolsa de iniciação científica pelo CNPq na UFRJ, mar/1988 a fev/1989
Bolsa de monitoria pela UFRJ, mar/1987 a nov/1987
Grant de pesquisa associado à bolsa de produtividade em pesquisa do CNPq, nível 1D, de agosto de 2003 até fevereiro de 2022 (R$1000,00 por mês)
Visita científica para participar, como Senior Fellow, de programa temático de três meses no IPAM/UCLA, sobre Mathematics of Turbulence, de 8 de setembro a 12 de dezembro de 2014
Visita científica à Université de Paris-Sud, Orsay, França - Cooperação França-Brasil, CNRS-CNPq, 18 de março a 17 de maio de 2002
Visita científica à Indiana University, IN, EUA, e à Texas A&M, TX, EUA - Financiada pelo ISC-Indiana University, Texas A&M e CNPq, 11 de janeiro a 11 de março de 2001
Prêmio Antônio Luís Vianna 1998-2000 para docentes recém-doutores - UFRJ/FUJB, auxílio de R$15.000,00, novembro de 2000
Visita científica ao à Indiana University, Bloomington, IN, EUA - Financiada pelo ISC-Indiana University, 30 de agosto a 30 de setembro de 2000
Visita científica à Indiana University, Bloomington, IN, EUA - Financiada pelo ISC-Indiana University, 1 de setembro de 1999 a 28 de abril de 2000
Programa ''Antônio Luís Vianna 1998-2000'' para docentes recém-doutores - Auxílio de R$5.000,00 para compra de materiais permanentes e de consumo, oferecido pela UFRJ/FUJB, outubro de 1998
Visita científica à Indiana University, Bloomington, IN, EUA - Financiada pela Indiana University e pela FAPERJ, 23 a 28 de agosto de 1998
Visita científica ao CNLS, Los Alamos National Lab, Los Alamos, NM, EUA - Financiada pelo CNLS e pela FAPERJ, 10 a 22 de agosto de 1998
Visita científica à Université de Paris-Sud, Orsay, França - Financiada pelo ISC-Indiana University, 2 de fevereiro a 3 de março de 1998
Primeiro lugar em concurso público para Professor Assistente do Departamento de Matemática Aplicada, IM-UFRJ, agosto de 1992
Primeiro lugar em concurso público para Professor Substituto do Departamento de Matemática Aplicada, IM-UFRJ, março de 1992
Membro do Comitê Editorial da Coleção Matemática Aplicada da SBM.
Representante no Colegiado do Programa de Pós-Graduação em Matemática do IM-UFRJ (2005 a 2012; 2016 a 2020; 2022 até o presente)
Participação em bancas de progressão funcional, estágio docente e concursos públicos
Referee de revistas especializadas, como Journal of Differential Equations, Discrete and Continuous Dynamical Systems, Indiana University Math Journal, SIAM Journal of Mathematical Analysis, SIAM Journal of Numerical Analysis, Journal of Mathematical Analysis and its Applications, Physica D, Nonlinear Analysis TMA, etc.
Consultor ad-hoc do CNPq, Capes e Faperj
Diretor Adjunto de Pós-Graduação do IM-UFRJ (novembro de 2007 a fevereiro de 2011)
Membro do Comitê de Área (Matemática/Probabilidade e Estatística) da CAPES (triênio 2011 a 2013)
Membro do Comitê de Área (Matemática/Probabilidade e Estatística) da CAPES (triênio 2008 a 2010)
Representante do CCMN no Comitê Institucional do PIBIC/UFRJ (2004 a março de 2011)
Coordenador do Programa de Pós-Graduação em Matemática Aplicada (2001 a 2007)
Representante do CCMN no Conselho de Ensino para Graduados (CEPG) da UFRJ (novembro de 2003 a outubro de 2005)
Membro da Câmara de Acompanhamento e Avaliação de Cursos de Pós-Graduação (CAAC) do CEPG-UFRJ (novembro de 2003 a outubro de 2005)
Participação em bancas de progressão funcional, estágio docente e concursos públicos
Escola de Verão do IM-UFRJ de 2007 (IM-UFRJ, Rio de Janeiro, janeiro a início de março de 2007 - Coordenador)
AMS-SBM Joint International Meeting (IMPA, Rio de Janeiro, 4 a 7 de junho de 2008 - Membro do Comitê Organizador Local)
II Workshop on Fluids and PDE (IM-UFRJ, Rio de Janeiro, 13 a 15 de agosto de 2008 - Coordenador)
1st Franco-Brazilian Fluids Summer School (Unicamp, Campinas, 13 a 22 de janeiro de 2010 - Membro do Comitê Científico)
III Workshop on Fluids and PDE (Unicamp, Campinas, 27 de junho a 1 de julho de 2011 - Membro do Comitê Científico)
IV Escola Brasileira de Equações Diferenciais (João Pessoa, Paraíba, 22 a 26 de agosto de 2011 - Membro do Comitê Científico)
Mini-simpósio MS0 Turbulence and Statistical Solutions in Incompressible Flows, 2011 SIAM Conference on Analysis of Partial Differential Equations (San Diego, California, 14 a 17 de novembro de 2011 - Membro do Comitê Organizador da Sessão)
Seminário de Fluidos (IM-UFRJ, Rio de Janeiro, 2013/1 - Coordenador)
Session 'Fluid Mechanics: from Turbulence to Free Boundaries', The Mathematical Congress of the Americas 2013 (Guajanuato, México, 5 a 9 de agosto de 2013 - Membro do Comitê Organizador da Sessão)
Conferência 5x05, Comemorando os 25 anos do Mestrado em Matemática Aplicada do IM-UFRJ (IM-UFRJ, Rio de Janeiro, 2 a 4 de abril de 2014 - Membro do Comitê Organizador)
IV Workshop on Fluids and PDE (IMPA, Rio de Janeiro, 26 a 30 de maio de 2014 - Membro do Comitê Científico)
V Workshop on Fluids and PDE (IMECC/Unicamp, Campinas, 20 de setembro a 1 de outubro de 2021 - Membro do Comitê Científico)
SciMLCon 2022 (Conferência virtual, 23 de março de 2022 - Membro do Comitê Organizador)
Em 16 de maio de 2018, participei do Festival Pint of Science Brasil. O Festival engloba eventos de divulgação científica em bares e restaurantes em diversas cidades do Brasil. Esta é a versão nacional do Pint of Science, realizado em cidades de diversos países. O Festival ocorre anualmente, sempre no mês de maio. A minha participação foi em uma mesa redonda com o Prof. Marcelo Viana (IMPA) e teve como tema "Dois chopes e senta que lá vem história: como a matemática está por trás da ciência e da arte da boa cerveja".
Também sou o autor do blog Cervejarte, onde falo sobre produção de cerveja, principalmente do ponto de vista de cerveja artesanal (caseira). Não é exatamente uma atividade de extensão, pois está mais para um hobby. No entanto, alguns artigos são mais técnicos, com detalhes matemáticos um pouco mais elaborados e apresentados de forma a serem inteligíveis para um público mais geral.
De Oliveira, M; Bertho, A. C. S.; Costa, B.; Somerlatte Silva, F.; Alves, M. B.; Ramos Ramirez, M.; Borges, R. B. R.; Marques, R.; Rosa, R. M. S.; Peregrino, R. L.; Lobo, V. G. R; Fonseca, T. C. O.. BR-EMS 2021 life table for the Brazilian insured population. Revista Brasileira de Estudos de População - REBEP, v. 40 (2023), p. 1–24. (DOI: 10.20947/s0102-3098a0252)
Becker, R. A.; Bercovici, H.; Biswas, A.; Cheskidov, A.; Constantin, P.; Eden, A.; Frazho, A.; Jolly, M.; Kukavica, I.; Pearcy, C.; Rosa, R. M. S.; Saut, J.-C.; Tannenbaum, A.; Temam, R.; Titi, E.; Voiculescu, D.. Remembrances of Ciprian Ilie Foias. American Mathematical Society. Notices, v. 69 (2022), p. 1529–1545. (DOI: 10.1090/noti2545)
Rosa, Ricardo M. S.; Temam, Roger M. Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations. Pure Appl. Funct. Anal. 7 (2022), no. 1, 327–355. (MR4396263)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 31 (2019), no. 3, 1689–1741. (DOI:10.1007/s10884-018-9719-2)
Cipolatti, R. A.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; On the existence, uniqueness and regularity of solutions of a viscoelastic Stokes problem modelling salt rocks, Appl. Math. Optim., 78 (2018), no. 2, 403–456. (DOI:10.1007/s00245-017-9411-7)
Cipolatti, R.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; A boundary value problem arising from nonlinear viscoelasticity: Mathematical analysis and numerical simulations, Appl. Math. Comput., 335 (2018), 237–247. (DOI:10.1016/j.amc.2018.04.034)
Bronzi, A. C.; Mondaini, C. F.; Rosa, R. M. S.; Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), no. 12, 8428–8484. (DOI:10.1016/j.jde.2016.02.027)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations, J. Stat. Phys., 160 (2015), no. 3, 519–531. (DOI:10.1007/s10955-015-1248-3)
Bronzi, Anne C.; Mondaini, Cecilia F.; Rosa, Ricardo M. S.; Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), no. 3, 1893–1921. (DOI:10.1137/130931631)
Bronzi, Anne; Rosa, Ricardo; On the convergence of statistical solutions of the 3D Navier-Stokes- model as vanishes, Discrete Contin. Dyn. Syst., 34 (2014), no. 1, 19–49. (DOI:10.3934/dcds.2014.34.19)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 63 (2013), no. 6, 2515–2573.
Foias, Ciprian; Rosa, Ricardo; Temam, Roger; Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1611–1631. (DOI:10.3934/dcds.2010.27.1611)
Balci, Nusret; Foias, Ciprian; Jolly, Michael S.; Rosa, Ricardo; On universal relations in 2-D turbulence, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1327–1351. (DOI:10.3934/dcds.2010.27.1327)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the stationary case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 5-6, 347–353. (DOI:10.1016/j.crma.2009.12.018)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the time-dependent case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 3-4, 235–240. (DOI:10.1016/j.crma.2009.12.017)
Rosa, Ricardo M. S.; Theory and applications of statistical solutions of the Navier-Stokes equations, in Partial differential equations and fluid mechanics, pp. 228–257, Cambridge Univ. Press, Cambridge, 2009.
Ramos, F.; Rosa, R.; Temam, R.; Statistical estimates for channel flows driven by a pressure gradient, Phys. D, 237 (2008), no. 10-12, 1368–1387. (DOI:10.1016/j.physd.2008.03.013)
Dieci, L.; Jolly, M. S.; Rosa, R.; Van Vleck, E. S.; Error in approximation of Lyapunov exponents on inertial manifolds: the Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), no. 3-4, 555–580. (DOI:10.3934/dcdsb.2008.9.555)
Rosa, Ricardo M. S.; Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), no. 1, 257–269. (DOI:10.1016/j.jde.2006.03.004)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Temam, R.; Kolmogorov theory via finite-time averages, Phys. D, 212 (2005), no. 3-4, 245–270. (DOI:10.1016/j.physd.2005.10.002)
Jolly, M. S.; Rosa, R.; Computation of non-smooth local centre manifolds, IMA J. Numer. Anal., 25 (2005), no. 4, 698–725. (DOI:10.1093/imanum/dri013)
Cabral, M.; Rosa, R.; Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), no. 3-4, 265–278. (DOI:10.1016/j.physd.2004.01.023)
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 473–496. (DOI:10.3934/dcds.2004.10.473)
Cabral, Marco; Rosa, Ricardo; Temam, Roger; Existence and dimension of the attractor for the Bénard problem on channel-like domains, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 89–116.
Rosa, Ricardo; Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dynam. Differential Equations, 15 (2003), no. 1, 61–86. (DOI:10.1023/A:1026153311546)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; On the Landau-Lifschitz degrees of freedom in 2-D turbulence, J. Statist. Phys., 111 (2003), no. 3-4, 1017–1019. (DOI:10.1023/A:1022814702548)
Rosa, Ricardo M. S.; Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence, Appl. Math., 47 (2002), no. 6, 485–516. (DOI:10.1023/A:1023297721804)
Goubet, Olivier; Rosa, Ricardo M. S.; Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), no. 1, 25–53. (DOI:10.1006/jdeq.2001.4163)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence, J. Statist. Phys., 108 (2002), no. 3-4, 591–645. (DOI:10.1023/A:1015782025005)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Estimates for the energy cascade in three-dimensional turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), no. 5, 499–504. (DOI:10.1016/S0764-4442(01)02008-0)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Cascade of energy in turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), no. 6, 509–514. (DOI:10.1016/S0764-4442(01)01831-6)
Jolly, M. S.; Rosa, R.; Temam, R.; Accurate computations on inertial manifolds, SIAM J. Sci. Comput., 22 (2000), no. 6, 2216–2238. (DOI:10.1137/S1064827599351738)
Rosa, Ricardo; The global attractor of a weakly damped, forced Korteweg-de Vries equation in , Mat. Contemp., 19 (2000), 129–152.
Jolly, M. S.; Rosa, R.; Temam, R.; Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation, Adv. Differential Equations, 5 (2000), no. 1-3, 31–66.
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), no. 5, 1369–1393. (DOI:10.1088/0951-7715/11/5/012)
Rosa, Ricardo; The global attractor for the D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), no. 1, 71–85. (DOI:10.1016/S0362-546X(97)00453-7)
Rosa, Ricardo; Temam, Roger; Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of computational mathematics (Rio de Janeiro, 1997), pp. 382–391, Springer, Berlin, 1997.
Moise, I.; Rosa, R.; On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Adv. Differential Equations, 2 (1997), no. 2, 257–296.
Rosa, Ricardo; Temam, Roger; Inertial manifolds and normal hyperbolicity, Acta Appl. Math., 45 (1996), no. 1, 1–50. (DOI:10.1007/BF00047882)
Castañeda, Nelson; Rosa, Ricardo; Optimal estimates for the uncoupling of differential equations, J. Dynam. Differential Equations, 8 (1996), no. 1, 103–139. (DOI:10.1007/BF02218616)
Rosa, Ricardo; Approximate inertial manifolds of exponential order, Discrete Contin. Dynam. Systems, 1 (1995), no. 3, 421–448. (DOI:10.3934/dcds.1995.1.421)
Rosa, Ricardo; Conjugacy of strongly continuous semigroups generated by normal operators, J. Dynam. Differential Equations, 7 (1995), no. 3, 471–490. (DOI:10.1007/BF02219373)
Foias, C.; Manley, O.; Rosa, R.; Temam, R.; Navier-Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001.
Rosa, Ricardo Martins da Silva; Attractors for weakly dissipative equations. Inertial manifolds and normal hyperbolicity. Approximate inertial manifolds of exponential order, Thesis (Ph.D.)–Indiana University, ProQuest LLC, Ann Arbor, MI, 1996.
Destaque Acadêmico do Instituto de Matemática, CCMN-UFRJ, 2009
Prêmio "Antônio Luís Vianna 2000" para docentes recém-doutores - UFRJ/FUJB, Novembro de 2000
Programa "Antônio Luís Vianna 1998" para docentes recém-doutores - UFRJ/FUJB, Outubro de 1998
John Ewing Book Award 1996 - Indiana University (US$300 em livros da Springer-Verlag)
Bolsa de Produtividade em Pesquisa, nível 1, pelo CNPq, ago/2003 até fev/2022
Bolsa de Produtividade em Pesquisa, nível 2B, pelo CNPq, ago/2001 a jul/2003
Bolsa de Produtividade em Pesquisa, nível 2B, pelo CNPq, ago/1999 a jul/2001
Bolsa de Produtividade em Pesquisa, nível 2C, pelo CNPq, ago/1997 a jul/1999
Bolsa de doutorado pelo CNPq na Indiana University, EUA, set/1992 a dez/1996
Bolsa de mestrado pelo CNPq na UFRJ, set/1989 a fev/1992
Bolsa de iniciação científica pelo CNPq na UFRJ, mar/1988 a fev/1989
Bolsa de monitoria pela UFRJ, mar/1987 a nov/1987
Grant de pesquisa associado à bolsa de produtividade em pesquisa do CNPq, nível 1D, de agosto de 2003 até fevereiro de 2022 (R$1000,00 por mês)
Visita científica para participar, como Senior Fellow, de programa temático de três meses no IPAM/UCLA, sobre Mathematics of Turbulence, de 8 de setembro a 12 de dezembro de 2014
Visita científica à Université de Paris-Sud, Orsay, França - Cooperação França-Brasil, CNRS-CNPq, 18 de março a 17 de maio de 2002
Visita científica à Indiana University, IN, EUA, e à Texas A&M, TX, EUA - Financiada pelo ISC-Indiana University, Texas A&M e CNPq, 11 de janeiro a 11 de março de 2001
Prêmio Antônio Luís Vianna 1998-2000 para docentes recém-doutores - UFRJ/FUJB, auxílio de R$15.000,00, novembro de 2000
Visita científica ao à Indiana University, Bloomington, IN, EUA - Financiada pelo ISC-Indiana University, 30 de agosto a 30 de setembro de 2000
Visita científica à Indiana University, Bloomington, IN, EUA - Financiada pelo ISC-Indiana University, 1 de setembro de 1999 a 28 de abril de 2000
Programa ''Antônio Luís Vianna 1998-2000'' para docentes recém-doutores - Auxílio de R$5.000,00 para compra de materiais permanentes e de consumo, oferecido pela UFRJ/FUJB, outubro de 1998
Visita científica à Indiana University, Bloomington, IN, EUA - Financiada pela Indiana University e pela FAPERJ, 23 a 28 de agosto de 1998
Visita científica ao CNLS, Los Alamos National Lab, Los Alamos, NM, EUA - Financiada pelo CNLS e pela FAPERJ, 10 a 22 de agosto de 1998
Visita científica à Université de Paris-Sud, Orsay, França - Financiada pelo ISC-Indiana University, 2 de fevereiro a 3 de março de 1998
Primeiro lugar em concurso público para Professor Assistente do Departamento de Matemática Aplicada, IM-UFRJ, agosto de 1992
Primeiro lugar em concurso público para Professor Substituto do Departamento de Matemática Aplicada, IM-UFRJ, março de 1992
Membro do Comitê Editorial da Coleção Matemática Aplicada da SBM.
Representante no Colegiado do Programa de Pós-Graduação em Matemática do IM-UFRJ (2005 a 2012; 2016 a 2020; 2022 até o presente)
Participação em bancas de progressão funcional, estágio docente e concursos públicos
Referee de revistas especializadas, como Journal of Differential Equations, Discrete and Continuous Dynamical Systems, Indiana University Math Journal, SIAM Journal of Mathematical Analysis, SIAM Journal of Numerical Analysis, Journal of Mathematical Analysis and its Applications, Physica D, Nonlinear Analysis TMA, etc.
Consultor ad-hoc do CNPq, Capes e Faperj
Diretor Adjunto de Pós-Graduação do IM-UFRJ (novembro de 2007 a fevereiro de 2011)
Membro do Comitê de Área (Matemática/Probabilidade e Estatística) da CAPES (triênio 2011 a 2013)
Membro do Comitê de Área (Matemática/Probabilidade e Estatística) da CAPES (triênio 2008 a 2010)
Representante do CCMN no Comitê Institucional do PIBIC/UFRJ (2004 a março de 2011)
Coordenador do Programa de Pós-Graduação em Matemática Aplicada (2001 a 2007)
Representante do CCMN no Conselho de Ensino para Graduados (CEPG) da UFRJ (novembro de 2003 a outubro de 2005)
Membro da Câmara de Acompanhamento e Avaliação de Cursos de Pós-Graduação (CAAC) do CEPG-UFRJ (novembro de 2003 a outubro de 2005)
Participação em bancas de progressão funcional, estágio docente e concursos públicos
Escola de Verão do IM-UFRJ de 2007 (IM-UFRJ, Rio de Janeiro, janeiro a início de março de 2007 - Coordenador)
AMS-SBM Joint International Meeting (IMPA, Rio de Janeiro, 4 a 7 de junho de 2008 - Membro do Comitê Organizador Local)
II Workshop on Fluids and PDE (IM-UFRJ, Rio de Janeiro, 13 a 15 de agosto de 2008 - Coordenador)
1st Franco-Brazilian Fluids Summer School (Unicamp, Campinas, 13 a 22 de janeiro de 2010 - Membro do Comitê Científico)
III Workshop on Fluids and PDE (Unicamp, Campinas, 27 de junho a 1 de julho de 2011 - Membro do Comitê Científico)
IV Escola Brasileira de Equações Diferenciais (João Pessoa, Paraíba, 22 a 26 de agosto de 2011 - Membro do Comitê Científico)
Mini-simpósio MS0 Turbulence and Statistical Solutions in Incompressible Flows, 2011 SIAM Conference on Analysis of Partial Differential Equations (San Diego, California, 14 a 17 de novembro de 2011 - Membro do Comitê Organizador da Sessão)
Seminário de Fluidos (IM-UFRJ, Rio de Janeiro, 2013/1 - Coordenador)
Session 'Fluid Mechanics: from Turbulence to Free Boundaries', The Mathematical Congress of the Americas 2013 (Guajanuato, México, 5 a 9 de agosto de 2013 - Membro do Comitê Organizador da Sessão)
Conferência 5x05, Comemorando os 25 anos do Mestrado em Matemática Aplicada do IM-UFRJ (IM-UFRJ, Rio de Janeiro, 2 a 4 de abril de 2014 - Membro do Comitê Organizador)
IV Workshop on Fluids and PDE (IMPA, Rio de Janeiro, 26 a 30 de maio de 2014 - Membro do Comitê Científico)
V Workshop on Fluids and PDE (IMECC/Unicamp, Campinas, 20 de setembro a 1 de outubro de 2021 - Membro do Comitê Científico)
SciMLCon 2022 (Conferência virtual, 23 de março de 2022 - Membro do Comitê Organizador)
Em 16 de maio de 2018, participei do Festival Pint of Science Brasil. O Festival engloba eventos de divulgação científica em bares e restaurantes em diversas cidades do Brasil. Esta é a versão nacional do Pint of Science, realizado em cidades de diversos países. O Festival ocorre anualmente, sempre no mês de maio. A minha participação foi em uma mesa redonda com o Prof. Marcelo Viana (IMPA) e teve como tema "Dois chopes e senta que lá vem história: como a matemática está por trás da ciência e da arte da boa cerveja".
Também sou o autor do blog Cervejarte, onde falo sobre produção de cerveja, principalmente do ponto de vista de cerveja artesanal (caseira). Não é exatamente uma atividade de extensão, pois está mais para um hobby. No entanto, alguns artigos são mais técnicos, com detalhes matemáticos um pouco mais elaborados e apresentados de forma a serem inteligíveis para um público mais geral.
De Oliveira, M; Bertho, A. C. S.; Costa, B.; Somerlatte Silva, F.; Alves, M. B.; Ramos Ramirez, M.; Borges, R. B. R.; Marques, R.; Rosa, R. M. S.; Peregrino, R. L.; Lobo, V. G. R; Fonseca, T. C. O.. BR-EMS 2021 life table for the Brazilian insured population. Revista Brasileira de Estudos de População - REBEP, v. 40 (2023), p. 1–24. (DOI: 10.20947/s0102-3098a0252)
Becker, R. A.; Bercovici, H.; Biswas, A.; Cheskidov, A.; Constantin, P.; Eden, A.; Frazho, A.; Jolly, M.; Kukavica, I.; Pearcy, C.; Rosa, R. M. S.; Saut, J.-C.; Tannenbaum, A.; Temam, R.; Titi, E.; Voiculescu, D.. Remembrances of Ciprian Ilie Foias. American Mathematical Society. Notices, v. 69 (2022), p. 1529–1545. (DOI: 10.1090/noti2545)
Rosa, Ricardo M. S.; Temam, Roger M. Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations. Pure Appl. Funct. Anal. 7 (2022), no. 1, 327–355. (MR4396263)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 31 (2019), no. 3, 1689–1741. (DOI:10.1007/s10884-018-9719-2)
Cipolatti, R. A.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; On the existence, uniqueness and regularity of solutions of a viscoelastic Stokes problem modelling salt rocks, Appl. Math. Optim., 78 (2018), no. 2, 403–456. (DOI:10.1007/s00245-017-9411-7)
Cipolatti, R.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; A boundary value problem arising from nonlinear viscoelasticity: Mathematical analysis and numerical simulations, Appl. Math. Comput., 335 (2018), 237–247. (DOI:10.1016/j.amc.2018.04.034)
Bronzi, A. C.; Mondaini, C. F.; Rosa, R. M. S.; Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), no. 12, 8428–8484. (DOI:10.1016/j.jde.2016.02.027)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations, J. Stat. Phys., 160 (2015), no. 3, 519–531. (DOI:10.1007/s10955-015-1248-3)
Bronzi, Anne C.; Mondaini, Cecilia F.; Rosa, Ricardo M. S.; Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), no. 3, 1893–1921. (DOI:10.1137/130931631)
Bronzi, Anne; Rosa, Ricardo; On the convergence of statistical solutions of the 3D Navier-Stokes- model as vanishes, Discrete Contin. Dyn. Syst., 34 (2014), no. 1, 19–49. (DOI:10.3934/dcds.2014.34.19)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 63 (2013), no. 6, 2515–2573.
Foias, Ciprian; Rosa, Ricardo; Temam, Roger; Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1611–1631. (DOI:10.3934/dcds.2010.27.1611)
Balci, Nusret; Foias, Ciprian; Jolly, Michael S.; Rosa, Ricardo; On universal relations in 2-D turbulence, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1327–1351. (DOI:10.3934/dcds.2010.27.1327)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the stationary case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 5-6, 347–353. (DOI:10.1016/j.crma.2009.12.018)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the time-dependent case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 3-4, 235–240. (DOI:10.1016/j.crma.2009.12.017)
Rosa, Ricardo M. S.; Theory and applications of statistical solutions of the Navier-Stokes equations, in Partial differential equations and fluid mechanics, pp. 228–257, Cambridge Univ. Press, Cambridge, 2009.
Ramos, F.; Rosa, R.; Temam, R.; Statistical estimates for channel flows driven by a pressure gradient, Phys. D, 237 (2008), no. 10-12, 1368–1387. (DOI:10.1016/j.physd.2008.03.013)
Dieci, L.; Jolly, M. S.; Rosa, R.; Van Vleck, E. S.; Error in approximation of Lyapunov exponents on inertial manifolds: the Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), no. 3-4, 555–580. (DOI:10.3934/dcdsb.2008.9.555)
Rosa, Ricardo M. S.; Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), no. 1, 257–269. (DOI:10.1016/j.jde.2006.03.004)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Temam, R.; Kolmogorov theory via finite-time averages, Phys. D, 212 (2005), no. 3-4, 245–270. (DOI:10.1016/j.physd.2005.10.002)
Jolly, M. S.; Rosa, R.; Computation of non-smooth local centre manifolds, IMA J. Numer. Anal., 25 (2005), no. 4, 698–725. (DOI:10.1093/imanum/dri013)
Cabral, M.; Rosa, R.; Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), no. 3-4, 265–278. (DOI:10.1016/j.physd.2004.01.023)
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 473–496. (DOI:10.3934/dcds.2004.10.473)
Cabral, Marco; Rosa, Ricardo; Temam, Roger; Existence and dimension of the attractor for the Bénard problem on channel-like domains, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 89–116.
Rosa, Ricardo; Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dynam. Differential Equations, 15 (2003), no. 1, 61–86. (DOI:10.1023/A:1026153311546)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; On the Landau-Lifschitz degrees of freedom in 2-D turbulence, J. Statist. Phys., 111 (2003), no. 3-4, 1017–1019. (DOI:10.1023/A:1022814702548)
Rosa, Ricardo M. S.; Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence, Appl. Math., 47 (2002), no. 6, 485–516. (DOI:10.1023/A:1023297721804)
Goubet, Olivier; Rosa, Ricardo M. S.; Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), no. 1, 25–53. (DOI:10.1006/jdeq.2001.4163)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence, J. Statist. Phys., 108 (2002), no. 3-4, 591–645. (DOI:10.1023/A:1015782025005)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Estimates for the energy cascade in three-dimensional turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), no. 5, 499–504. (DOI:10.1016/S0764-4442(01)02008-0)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Cascade of energy in turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), no. 6, 509–514. (DOI:10.1016/S0764-4442(01)01831-6)
Jolly, M. S.; Rosa, R.; Temam, R.; Accurate computations on inertial manifolds, SIAM J. Sci. Comput., 22 (2000), no. 6, 2216–2238. (DOI:10.1137/S1064827599351738)
Rosa, Ricardo; The global attractor of a weakly damped, forced Korteweg-de Vries equation in , Mat. Contemp., 19 (2000), 129–152.
Jolly, M. S.; Rosa, R.; Temam, R.; Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation, Adv. Differential Equations, 5 (2000), no. 1-3, 31–66.
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), no. 5, 1369–1393. (DOI:10.1088/0951-7715/11/5/012)
Rosa, Ricardo; The global attractor for the D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), no. 1, 71–85. (DOI:10.1016/S0362-546X(97)00453-7)
Rosa, Ricardo; Temam, Roger; Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of computational mathematics (Rio de Janeiro, 1997), pp. 382–391, Springer, Berlin, 1997.
Moise, I.; Rosa, R.; On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Adv. Differential Equations, 2 (1997), no. 2, 257–296.
Rosa, Ricardo; Temam, Roger; Inertial manifolds and normal hyperbolicity, Acta Appl. Math., 45 (1996), no. 1, 1–50. (DOI:10.1007/BF00047882)
Castañeda, Nelson; Rosa, Ricardo; Optimal estimates for the uncoupling of differential equations, J. Dynam. Differential Equations, 8 (1996), no. 1, 103–139. (DOI:10.1007/BF02218616)
Rosa, Ricardo; Approximate inertial manifolds of exponential order, Discrete Contin. Dynam. Systems, 1 (1995), no. 3, 421–448. (DOI:10.3934/dcds.1995.1.421)
Rosa, Ricardo; Conjugacy of strongly continuous semigroups generated by normal operators, J. Dynam. Differential Equations, 7 (1995), no. 3, 471–490. (DOI:10.1007/BF02219373)
Foias, C.; Manley, O.; Rosa, R.; Temam, R.; Navier-Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001.
Rosa, Ricardo Martins da Silva; Attractors for weakly dissipative equations. Inertial manifolds and normal hyperbolicity. Approximate inertial manifolds of exponential order, Thesis (Ph.D.)–Indiana University, ProQuest LLC, Ann Arbor, MI, 1996.
"Academic Distinction" at the Instituto de Matemática, CCMN-UFRJ, 2009
"Prize Antônio Luís Vianna 2000" for Professors with recent Graduate degree - UFRJ/FUJB, November 2000
"Program Antônio Luís Vianna 1998" for Professors with recent Graduate degree - UFRJ/FUJB, October 1998
"John Ewing Book Award 1996" - Indiana University (US$300 in books from Springer-Verlag)
CNPq Research Fellowship, level 1D, 2003 to 2022
CNPq Research Fellowship, level 2A, Aug/2001 to Jul/2003
CNPq Research Fellowship, level 2B, Aug/1999 to Jul/2001
CNPq Research Fellowship, level 2C, Aug/1997 to Jul/1999
PhD Scholarship from CNPq, Brasil, at the Indiana University, EUA, set/1992 a dez/1996
MSc Scholarship from CNPq, Brasil, at UFRJ, set/1989 a fev/1992
Undergraduate Scholarship from CNPq, Brasil, at UFRJ, mar/1988 a fev/1989
Undergraduate Scholarship from CNPq, Brasil, at UFRJ, mar/1987 a nov/1987
Scientific Visit to IPAM/UCLA to participate, as a Senior Fellow in the Mathematics of Turbulence thematic program, from September 8 to December 12, 2014
Research grant from CNPq as part of the research fellowship - From August 2003 to February 2022 (R$1,000.00 per month)
Scientific visit to the Université de Paris-Sud, Orsay, France - French-Brasil Cooperation, CNRS-CNPq - March 18 to May 17, 2002
Scientific visit to the Indiana University, IN, USA, and Texas A&M, TX, USA - Supported by the ISC-Indiana University, Texas A&M and CNPq - January 11 to March 11, 2001
Prize "Antônio Luís Vianna 1998-2000" - from UFRJ/FUJB - November 2000 (R$15,000.00 grant)
Scientific visit to the Indiana University, Bloomington, IN, USA - Supported by ISC-Indiana University - August 30 to September 39, 2000
Scientific visit to the Indiana University, Bloomington, IN, USA - Supported by the ISC-Indiana University - September 1, 1999 to April 28, 2000
Program "Antônio Luís Vianna 1998-2000" - from FUJB/UFRJ - October 1998 (R$5,000.00 grant)
Scientific visit to the Indiana University, Bloomington, IN, USA - Supported by the ISC-Indiana University and by FAPERJ, Rio de Janeiro - August 23 to 28, 1998
Scientific visit to the CNLS, Los Alamos National Lab, Los Alamos, NM, EUA - Supported by CNLS and FAPERJ - August 10 to 22, 1998
Scientific visit to the Université de Paris-Sud, Orsay, France - Supported by the ISC-Indiana University - February 2 to March 3, 1998
First place in the exam for a position as Assistant Professor at the Department of Applied Mathematicas, UFRJ, August 1992.
First place in the exam for a position as Substitute Professor at the Department of Applied Mathematicas, UFRJ, March 1992.
Member of the Editorial Committee of the Coleção Matemática Aplicada of the Brazilian Mathematical Society.
Representative in the Committee of the Mathematics Graduate Program at UFRJ (Programa de Pós-Graduação em Matemática/IM-UFRJ) (2005 to 2012; 2016 to 2020; and 2022 on)
Member of various examining committees for university admission, carreer progression, tenure, etc.
Referee of research journals like Journal of Differential Equations, Discrete and Continuous Dynamical Systems, Indiana University Math Journal, SIAM Journal of Mathematical Analysis, SIAM Journal of Numerical Analysis, Journal of Mathematical Analysis and its Applications, Physica D, Nonlinear Analysis TMA, etc.
Ad-hoc referee for the financial agencies CNPq, CAPES and FAPERJ
Associate Director for the Graduate Programs at the IM-UFRJ (November 2007 to February 2011)
Member of the CAPES National Committee in the Area of Mathematics/Probability and Statistics (2011 to 2013)
Member of the CAPES National Committee in the Area of Mathematics/Probability and Statistics (2008 to 2010)
Representative at the University Committee for the Undergraduate Research Program (Comitê Institucional do PIBIC/UFRJ) (March 2004 to March 2011)
Coordinator of the Graduate Programa in Applied Mathematics at the IM-UFRJ (2001 to 2007)
Representative in the Graduate University Council Conselho de Ensino para Graduados (CEPG) at UFRJ (November 2003 to October 2005)
Member of the Council for Evaluation and Follow-up of Graduate Programs (Câmara de Avaliação e Acompanhamento de Cursos) of the CEPG-UFRJ (November 2003 to October 2005)
Member of various examining committees for university admission, carreer progression, tenure, etc.
Summer School IM-UFRJ 2007 (IM-UFRJ, Rio de Janeiro, early January to early March - Coordinator)
AMS-SBM Joint International Meeting (IMPA, Rio de Janeiro, June 4-7, 2008 - Member of the Local Organizing Committee)
II Workshop on Fluids and PDE (IM-UFRJ, Rio de Janeiro, August 13-15, 2008 - Coordinator)
1st Franco-Brazilian Fluids Summer School (Unicamp, Campinas, January 13-22, 2010 - Member of the Scientific Committee)
III Workshop on Fluids and PDE (Unicamp, Campinas, June 27 to July 1, 2011 - Member of the Scientific Committee)
IV Escola Brasileira de Equações Diferenciais (João Pessoa, Paraíba, August 22-26, 2011 - Member of the Scientific Committee)
Mini-symposium MS0 Turbulence and Statistical Solutions in Incompressible Flows, 2011 SIAM Conference on Analysis of Partial Differential Equations (San Diego, California, November 14-17, 2011 - Member of the Organizing Committee of the Session)
Fluid Seminar at IM-UFRJ (IM-UFRJ, Rio de Janeiro, 2013/1 - Coordinator)
Session 'Fluid Mechanics: from Turbulence to Free Boundaries', The Mathematical Congress of the Americas 2013 (Guajanuato, México, August 5-9, 2013 - Member of the Organizing Committee of the Session)
Conference 5x05. Celebrating 25 years of the MSc program in Applied Mathematics of the IM-UFRJ (IM-UFRJ, Rio de Janeiro, April 2-4, 2014 - Member of the Local Organizing Committee)
IV Workshop on Fluids and PDE (IMPA, Rio de Janeiro, May 26-30, 2014 - Member of the Scientific Committee)
V Workshop on Fluids and PDE (IMECC/Unicamp, Campinas, September 20 to October 1, 2021 - Member of the Scientific Committee)
SciMLCon 2022 (Virtual Conference, March 23rd, 2022 - Member of the Organizing Committee)
In May 16, 2018, I participated in the Pint of Science Brasil, which is the Brazilian version of the Pint of Science. This is a worldwide Festival in which Science is brought to the main public through discussions in selected bars and restaurants in the participating cities. The festival occurs once a year, always in May. My participation was together with Prof. Marcelo Viana (IMPA), on how Mathematics is behind the Science and Art of a good beer: "Dois chopes e senta que lá vem história: como a matemática está por trás da ciência e da arte da boa cerveja".
I am also the author of the blog Cervejarte, on which I write about beer production, particularly from the homebrewing perspective. It is more a hobby than a Mathematics outreach activity, but some of the posts are more technical, with more involved mathematical details, and which are presented in a form accessible to a more general audience.
De Oliveira, M; Bertho, A. C. S.; Costa, B.; Somerlatte Silva, F.; Alves, M. B.; Ramos Ramirez, M.; Borges, R. B. R.; Marques, R.; Rosa, R. M. S.; Peregrino, R. L.; Lobo, V. G. R; Fonseca, T. C. O.. BR-EMS 2021 life table for the Brazilian insured population. Revista Brasileira de Estudos de População - REBEP, v. 40 (2023), p. 1–24. (DOI: 10.20947/s0102-3098a0252)
Becker, R. A.; Bercovici, H.; Biswas, A.; Cheskidov, A.; Constantin, P.; Eden, A.; Frazho, A.; Jolly, M.; Kukavica, I.; Pearcy, C.; Rosa, R. M. S.; Saut, J.-C.; Tannenbaum, A.; Temam, R.; Titi, E.; Voiculescu, D.. Remembrances of Ciprian Ilie Foias. American Mathematical Society. Notices, v. 69 (2022), p. 1529–1545. (DOI: 10.1090/noti2545)
Rosa, Ricardo M. S.; Temam, Roger M. Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations. Pure Appl. Funct. Anal. 7 (2022), no. 1, 327–355. (MR4396263)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 31 (2019), no. 3, 1689–1741. (DOI:10.1007/s10884-018-9719-2)
Cipolatti, R. A.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; On the existence, uniqueness and regularity of solutions of a viscoelastic Stokes problem modelling salt rocks, Appl. Math. Optim., 78 (2018), no. 2, 403–456. (DOI:10.1007/s00245-017-9411-7)
Cipolatti, R.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; A boundary value problem arising from nonlinear viscoelasticity: Mathematical analysis and numerical simulations, Appl. Math. Comput., 335 (2018), 237–247. (DOI:10.1016/j.amc.2018.04.034)
Bronzi, A. C.; Mondaini, C. F.; Rosa, R. M. S.; Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), no. 12, 8428–8484. (DOI:10.1016/j.jde.2016.02.027)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations, J. Stat. Phys., 160 (2015), no. 3, 519–531. (DOI:10.1007/s10955-015-1248-3)
Bronzi, Anne C.; Mondaini, Cecilia F.; Rosa, Ricardo M. S.; Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), no. 3, 1893–1921. (DOI:10.1137/130931631)
Bronzi, Anne; Rosa, Ricardo; On the convergence of statistical solutions of the 3D Navier-Stokes- model as vanishes, Discrete Contin. Dyn. Syst., 34 (2014), no. 1, 19–49. (DOI:10.3934/dcds.2014.34.19)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 63 (2013), no. 6, 2515–2573.
Foias, Ciprian; Rosa, Ricardo; Temam, Roger; Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1611–1631. (DOI:10.3934/dcds.2010.27.1611)
Balci, Nusret; Foias, Ciprian; Jolly, Michael S.; Rosa, Ricardo; On universal relations in 2-D turbulence, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1327–1351. (DOI:10.3934/dcds.2010.27.1327)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the stationary case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 5-6, 347–353. (DOI:10.1016/j.crma.2009.12.018)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the time-dependent case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 3-4, 235–240. (DOI:10.1016/j.crma.2009.12.017)
Rosa, Ricardo M. S.; Theory and applications of statistical solutions of the Navier-Stokes equations, in Partial differential equations and fluid mechanics, pp. 228–257, Cambridge Univ. Press, Cambridge, 2009.
Ramos, F.; Rosa, R.; Temam, R.; Statistical estimates for channel flows driven by a pressure gradient, Phys. D, 237 (2008), no. 10-12, 1368–1387. (DOI:10.1016/j.physd.2008.03.013)
Dieci, L.; Jolly, M. S.; Rosa, R.; Van Vleck, E. S.; Error in approximation of Lyapunov exponents on inertial manifolds: the Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), no. 3-4, 555–580. (DOI:10.3934/dcdsb.2008.9.555)
Rosa, Ricardo M. S.; Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), no. 1, 257–269. (DOI:10.1016/j.jde.2006.03.004)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Temam, R.; Kolmogorov theory via finite-time averages, Phys. D, 212 (2005), no. 3-4, 245–270. (DOI:10.1016/j.physd.2005.10.002)
Jolly, M. S.; Rosa, R.; Computation of non-smooth local centre manifolds, IMA J. Numer. Anal., 25 (2005), no. 4, 698–725. (DOI:10.1093/imanum/dri013)
Cabral, M.; Rosa, R.; Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), no. 3-4, 265–278. (DOI:10.1016/j.physd.2004.01.023)
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 473–496. (DOI:10.3934/dcds.2004.10.473)
Cabral, Marco; Rosa, Ricardo; Temam, Roger; Existence and dimension of the attractor for the Bénard problem on channel-like domains, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 89–116.
Rosa, Ricardo; Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dynam. Differential Equations, 15 (2003), no. 1, 61–86. (DOI:10.1023/A:1026153311546)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; On the Landau-Lifschitz degrees of freedom in 2-D turbulence, J. Statist. Phys., 111 (2003), no. 3-4, 1017–1019. (DOI:10.1023/A:1022814702548)
Rosa, Ricardo M. S.; Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence, Appl. Math., 47 (2002), no. 6, 485–516. (DOI:10.1023/A:1023297721804)
Goubet, Olivier; Rosa, Ricardo M. S.; Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), no. 1, 25–53. (DOI:10.1006/jdeq.2001.4163)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence, J. Statist. Phys., 108 (2002), no. 3-4, 591–645. (DOI:10.1023/A:1015782025005)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Estimates for the energy cascade in three-dimensional turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), no. 5, 499–504. (DOI:10.1016/S0764-4442(01)02008-0)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Cascade of energy in turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), no. 6, 509–514. (DOI:10.1016/S0764-4442(01)01831-6)
Jolly, M. S.; Rosa, R.; Temam, R.; Accurate computations on inertial manifolds, SIAM J. Sci. Comput., 22 (2000), no. 6, 2216–2238. (DOI:10.1137/S1064827599351738)
Rosa, Ricardo; The global attractor of a weakly damped, forced Korteweg-de Vries equation in , Mat. Contemp., 19 (2000), 129–152.
Jolly, M. S.; Rosa, R.; Temam, R.; Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation, Adv. Differential Equations, 5 (2000), no. 1-3, 31–66.
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), no. 5, 1369–1393. (DOI:10.1088/0951-7715/11/5/012)
Rosa, Ricardo; The global attractor for the D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), no. 1, 71–85. (DOI:10.1016/S0362-546X(97)00453-7)
Rosa, Ricardo; Temam, Roger; Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of computational mathematics (Rio de Janeiro, 1997), pp. 382–391, Springer, Berlin, 1997.
Moise, I.; Rosa, R.; On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Adv. Differential Equations, 2 (1997), no. 2, 257–296.
Rosa, Ricardo; Temam, Roger; Inertial manifolds and normal hyperbolicity, Acta Appl. Math., 45 (1996), no. 1, 1–50. (DOI:10.1007/BF00047882)
Castañeda, Nelson; Rosa, Ricardo; Optimal estimates for the uncoupling of differential equations, J. Dynam. Differential Equations, 8 (1996), no. 1, 103–139. (DOI:10.1007/BF02218616)
Rosa, Ricardo; Approximate inertial manifolds of exponential order, Discrete Contin. Dynam. Systems, 1 (1995), no. 3, 421–448. (DOI:10.3934/dcds.1995.1.421)
Rosa, Ricardo; Conjugacy of strongly continuous semigroups generated by normal operators, J. Dynam. Differential Equations, 7 (1995), no. 3, 471–490. (DOI:10.1007/BF02219373)
Foias, C.; Manley, O.; Rosa, R.; Temam, R.; Navier-Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001.
Rosa, Ricardo Martins da Silva; Attractors for weakly dissipative equations. Inertial manifolds and normal hyperbolicity. Approximate inertial manifolds of exponential order, Thesis (Ph.D.)–Indiana University, ProQuest LLC, Ann Arbor, MI, 1996.
"Academic Distinction" at the Instituto de Matemática, CCMN-UFRJ, 2009
"Prize Antônio Luís Vianna 2000" for Professors with recent Graduate degree - UFRJ/FUJB, November 2000
"Program Antônio Luís Vianna 1998" for Professors with recent Graduate degree - UFRJ/FUJB, October 1998
"John Ewing Book Award 1996" - Indiana University (US$300 in books from Springer-Verlag)
CNPq Research Fellowship, level 1D, 2003 to 2022
CNPq Research Fellowship, level 2A, Aug/2001 to Jul/2003
CNPq Research Fellowship, level 2B, Aug/1999 to Jul/2001
CNPq Research Fellowship, level 2C, Aug/1997 to Jul/1999
PhD Scholarship from CNPq, Brasil, at the Indiana University, EUA, set/1992 a dez/1996
MSc Scholarship from CNPq, Brasil, at UFRJ, set/1989 a fev/1992
Undergraduate Scholarship from CNPq, Brasil, at UFRJ, mar/1988 a fev/1989
Undergraduate Scholarship from CNPq, Brasil, at UFRJ, mar/1987 a nov/1987
Scientific Visit to IPAM/UCLA to participate, as a Senior Fellow in the Mathematics of Turbulence thematic program, from September 8 to December 12, 2014
Research grant from CNPq as part of the research fellowship - From August 2003 to February 2022 (R$1,000.00 per month)
Scientific visit to the Université de Paris-Sud, Orsay, France - French-Brasil Cooperation, CNRS-CNPq - March 18 to May 17, 2002
Scientific visit to the Indiana University, IN, USA, and Texas A&M, TX, USA - Supported by the ISC-Indiana University, Texas A&M and CNPq - January 11 to March 11, 2001
Prize "Antônio Luís Vianna 1998-2000" - from UFRJ/FUJB - November 2000 (R$15,000.00 grant)
Scientific visit to the Indiana University, Bloomington, IN, USA - Supported by ISC-Indiana University - August 30 to September 39, 2000
Scientific visit to the Indiana University, Bloomington, IN, USA - Supported by the ISC-Indiana University - September 1, 1999 to April 28, 2000
Program "Antônio Luís Vianna 1998-2000" - from FUJB/UFRJ - October 1998 (R$5,000.00 grant)
Scientific visit to the Indiana University, Bloomington, IN, USA - Supported by the ISC-Indiana University and by FAPERJ, Rio de Janeiro - August 23 to 28, 1998
Scientific visit to the CNLS, Los Alamos National Lab, Los Alamos, NM, EUA - Supported by CNLS and FAPERJ - August 10 to 22, 1998
Scientific visit to the Université de Paris-Sud, Orsay, France - Supported by the ISC-Indiana University - February 2 to March 3, 1998
First place in the exam for a position as Assistant Professor at the Department of Applied Mathematicas, UFRJ, August 1992.
First place in the exam for a position as Substitute Professor at the Department of Applied Mathematicas, UFRJ, March 1992.
Member of the Editorial Committee of the Coleção Matemática Aplicada of the Brazilian Mathematical Society.
Representative in the Committee of the Mathematics Graduate Program at UFRJ (Programa de Pós-Graduação em Matemática/IM-UFRJ) (2005 to 2012; 2016 to 2020; and 2022 on)
Member of various examining committees for university admission, carreer progression, tenure, etc.
Referee of research journals like Journal of Differential Equations, Discrete and Continuous Dynamical Systems, Indiana University Math Journal, SIAM Journal of Mathematical Analysis, SIAM Journal of Numerical Analysis, Journal of Mathematical Analysis and its Applications, Physica D, Nonlinear Analysis TMA, etc.
Ad-hoc referee for the financial agencies CNPq, CAPES and FAPERJ
Associate Director for the Graduate Programs at the IM-UFRJ (November 2007 to February 2011)
Member of the CAPES National Committee in the Area of Mathematics/Probability and Statistics (2011 to 2013)
Member of the CAPES National Committee in the Area of Mathematics/Probability and Statistics (2008 to 2010)
Representative at the University Committee for the Undergraduate Research Program (Comitê Institucional do PIBIC/UFRJ) (March 2004 to March 2011)
Coordinator of the Graduate Programa in Applied Mathematics at the IM-UFRJ (2001 to 2007)
Representative in the Graduate University Council Conselho de Ensino para Graduados (CEPG) at UFRJ (November 2003 to October 2005)
Member of the Council for Evaluation and Follow-up of Graduate Programs (Câmara de Avaliação e Acompanhamento de Cursos) of the CEPG-UFRJ (November 2003 to October 2005)
Member of various examining committees for university admission, carreer progression, tenure, etc.
Summer School IM-UFRJ 2007 (IM-UFRJ, Rio de Janeiro, early January to early March - Coordinator)
AMS-SBM Joint International Meeting (IMPA, Rio de Janeiro, June 4-7, 2008 - Member of the Local Organizing Committee)
II Workshop on Fluids and PDE (IM-UFRJ, Rio de Janeiro, August 13-15, 2008 - Coordinator)
1st Franco-Brazilian Fluids Summer School (Unicamp, Campinas, January 13-22, 2010 - Member of the Scientific Committee)
III Workshop on Fluids and PDE (Unicamp, Campinas, June 27 to July 1, 2011 - Member of the Scientific Committee)
IV Escola Brasileira de Equações Diferenciais (João Pessoa, Paraíba, August 22-26, 2011 - Member of the Scientific Committee)
Mini-symposium MS0 Turbulence and Statistical Solutions in Incompressible Flows, 2011 SIAM Conference on Analysis of Partial Differential Equations (San Diego, California, November 14-17, 2011 - Member of the Organizing Committee of the Session)
Fluid Seminar at IM-UFRJ (IM-UFRJ, Rio de Janeiro, 2013/1 - Coordinator)
Session 'Fluid Mechanics: from Turbulence to Free Boundaries', The Mathematical Congress of the Americas 2013 (Guajanuato, México, August 5-9, 2013 - Member of the Organizing Committee of the Session)
Conference 5x05. Celebrating 25 years of the MSc program in Applied Mathematics of the IM-UFRJ (IM-UFRJ, Rio de Janeiro, April 2-4, 2014 - Member of the Local Organizing Committee)
IV Workshop on Fluids and PDE (IMPA, Rio de Janeiro, May 26-30, 2014 - Member of the Scientific Committee)
V Workshop on Fluids and PDE (IMECC/Unicamp, Campinas, September 20 to October 1, 2021 - Member of the Scientific Committee)
SciMLCon 2022 (Virtual Conference, March 23rd, 2022 - Member of the Organizing Committee)
In May 16, 2018, I participated in the Pint of Science Brasil, which is the Brazilian version of the Pint of Science. This is a worldwide Festival in which Science is brought to the main public through discussions in selected bars and restaurants in the participating cities. The festival occurs once a year, always in May. My participation was together with Prof. Marcelo Viana (IMPA), on how Mathematics is behind the Science and Art of a good beer: "Dois chopes e senta que lá vem história: como a matemática está por trás da ciência e da arte da boa cerveja".
I am also the author of the blog Cervejarte, on which I write about beer production, particularly from the homebrewing perspective. It is more a hobby than a Mathematics outreach activity, but some of the posts are more technical, with more involved mathematical details, and which are presented in a form accessible to a more general audience.
De Oliveira, M; Bertho, A. C. S.; Costa, B.; Somerlatte Silva, F.; Alves, M. B.; Ramos Ramirez, M.; Borges, R. B. R.; Marques, R.; Rosa, R. M. S.; Peregrino, R. L.; Lobo, V. G. R; Fonseca, T. C. O.. BR-EMS 2021 life table for the Brazilian insured population. Revista Brasileira de Estudos de População - REBEP, v. 40 (2023), p. 1–24. (DOI: 10.20947/s0102-3098a0252)
Becker, R. A.; Bercovici, H.; Biswas, A.; Cheskidov, A.; Constantin, P.; Eden, A.; Frazho, A.; Jolly, M.; Kukavica, I.; Pearcy, C.; Rosa, R. M. S.; Saut, J.-C.; Tannenbaum, A.; Temam, R.; Titi, E.; Voiculescu, D.. Remembrances of Ciprian Ilie Foias. American Mathematical Society. Notices, v. 69 (2022), p. 1529–1545. (DOI: 10.1090/noti2545)
Rosa, Ricardo M. S.; Temam, Roger M. Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations. Pure Appl. Funct. Anal. 7 (2022), no. 1, 327–355. (MR4396263)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 31 (2019), no. 3, 1689–1741. (DOI:10.1007/s10884-018-9719-2)
Cipolatti, R. A.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; On the existence, uniqueness and regularity of solutions of a viscoelastic Stokes problem modelling salt rocks, Appl. Math. Optim., 78 (2018), no. 2, 403–456. (DOI:10.1007/s00245-017-9411-7)
Cipolatti, R.; Liu, I.-S.; Palermo, L. A.; Rincon, M. A.; Rosa, R. M. S.; A boundary value problem arising from nonlinear viscoelasticity: Mathematical analysis and numerical simulations, Appl. Math. Comput., 335 (2018), 237–247. (DOI:10.1016/j.amc.2018.04.034)
Bronzi, A. C.; Mondaini, C. F.; Rosa, R. M. S.; Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), no. 12, 8428–8484. (DOI:10.1016/j.jde.2016.02.027)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.; Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations, J. Stat. Phys., 160 (2015), no. 3, 519–531. (DOI:10.1007/s10955-015-1248-3)
Bronzi, Anne C.; Mondaini, Cecilia F.; Rosa, Ricardo M. S.; Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46 (2014), no. 3, 1893–1921. (DOI:10.1137/130931631)
Bronzi, Anne; Rosa, Ricardo; On the convergence of statistical solutions of the 3D Navier-Stokes- model as vanishes, Discrete Contin. Dyn. Syst., 34 (2014), no. 1, 19–49. (DOI:10.3934/dcds.2014.34.19)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 63 (2013), no. 6, 2515–2573.
Foias, Ciprian; Rosa, Ricardo; Temam, Roger; Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1611–1631. (DOI:10.3934/dcds.2010.27.1611)
Balci, Nusret; Foias, Ciprian; Jolly, Michael S.; Rosa, Ricardo; On universal relations in 2-D turbulence, Discrete Contin. Dyn. Syst., 27 (2010), no. 4, 1327–1351. (DOI:10.3934/dcds.2010.27.1327)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the stationary case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 5-6, 347–353. (DOI:10.1016/j.crma.2009.12.018)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger; A note on statistical solutions of the three-dimensional Navier-Stokes equations: the time-dependent case, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 3-4, 235–240. (DOI:10.1016/j.crma.2009.12.017)
Rosa, Ricardo M. S.; Theory and applications of statistical solutions of the Navier-Stokes equations, in Partial differential equations and fluid mechanics, pp. 228–257, Cambridge Univ. Press, Cambridge, 2009.
Ramos, F.; Rosa, R.; Temam, R.; Statistical estimates for channel flows driven by a pressure gradient, Phys. D, 237 (2008), no. 10-12, 1368–1387. (DOI:10.1016/j.physd.2008.03.013)
Dieci, L.; Jolly, M. S.; Rosa, R.; Van Vleck, E. S.; Error in approximation of Lyapunov exponents on inertial manifolds: the Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), no. 3-4, 555–580. (DOI:10.3934/dcdsb.2008.9.555)
Rosa, Ricardo M. S.; Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), no. 1, 257–269. (DOI:10.1016/j.jde.2006.03.004)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Temam, R.; Kolmogorov theory via finite-time averages, Phys. D, 212 (2005), no. 3-4, 245–270. (DOI:10.1016/j.physd.2005.10.002)
Jolly, M. S.; Rosa, R.; Computation of non-smooth local centre manifolds, IMA J. Numer. Anal., 25 (2005), no. 4, 698–725. (DOI:10.1093/imanum/dri013)
Cabral, M.; Rosa, R.; Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), no. 3-4, 265–278. (DOI:10.1016/j.physd.2004.01.023)
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 473–496. (DOI:10.3934/dcds.2004.10.473)
Cabral, Marco; Rosa, Ricardo; Temam, Roger; Existence and dimension of the attractor for the Bénard problem on channel-like domains, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 89–116.
Rosa, Ricardo; Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dynam. Differential Equations, 15 (2003), no. 1, 61–86. (DOI:10.1023/A:1026153311546)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; On the Landau-Lifschitz degrees of freedom in 2-D turbulence, J. Statist. Phys., 111 (2003), no. 3-4, 1017–1019. (DOI:10.1023/A:1022814702548)
Rosa, Ricardo M. S.; Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence, Appl. Math., 47 (2002), no. 6, 485–516. (DOI:10.1023/A:1023297721804)
Goubet, Olivier; Rosa, Ricardo M. S.; Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), no. 1, 25–53. (DOI:10.1006/jdeq.2001.4163)
Foias, C.; Jolly, M. S.; Manley, O. P.; Rosa, R.; Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence, J. Statist. Phys., 108 (2002), no. 3-4, 591–645. (DOI:10.1023/A:1015782025005)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Estimates for the energy cascade in three-dimensional turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), no. 5, 499–504. (DOI:10.1016/S0764-4442(01)02008-0)
Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger; Cascade of energy in turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), no. 6, 509–514. (DOI:10.1016/S0764-4442(01)01831-6)
Jolly, M. S.; Rosa, R.; Temam, R.; Accurate computations on inertial manifolds, SIAM J. Sci. Comput., 22 (2000), no. 6, 2216–2238. (DOI:10.1137/S1064827599351738)
Rosa, Ricardo; The global attractor of a weakly damped, forced Korteweg-de Vries equation in , Mat. Contemp., 19 (2000), 129–152.
Jolly, M. S.; Rosa, R.; Temam, R.; Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation, Adv. Differential Equations, 5 (2000), no. 1-3, 31–66.
Moise, Ioana; Rosa, Ricardo; Wang, Xiaoming; Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), no. 5, 1369–1393. (DOI:10.1088/0951-7715/11/5/012)
Rosa, Ricardo; The global attractor for the D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), no. 1, 71–85. (DOI:10.1016/S0362-546X(97)00453-7)
Rosa, Ricardo; Temam, Roger; Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of computational mathematics (Rio de Janeiro, 1997), pp. 382–391, Springer, Berlin, 1997.
Moise, I.; Rosa, R.; On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Adv. Differential Equations, 2 (1997), no. 2, 257–296.
Rosa, Ricardo; Temam, Roger; Inertial manifolds and normal hyperbolicity, Acta Appl. Math., 45 (1996), no. 1, 1–50. (DOI:10.1007/BF00047882)
Castañeda, Nelson; Rosa, Ricardo; Optimal estimates for the uncoupling of differential equations, J. Dynam. Differential Equations, 8 (1996), no. 1, 103–139. (DOI:10.1007/BF02218616)
Rosa, Ricardo; Approximate inertial manifolds of exponential order, Discrete Contin. Dynam. Systems, 1 (1995), no. 3, 421–448. (DOI:10.3934/dcds.1995.1.421)
Rosa, Ricardo; Conjugacy of strongly continuous semigroups generated by normal operators, J. Dynam. Differential Equations, 7 (1995), no. 3, 471–490. (DOI:10.1007/BF02219373)
Foias, C.; Manley, O.; Rosa, R.; Temam, R.; Navier-Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001.
Rosa, Ricardo Martins da Silva; Attractors for weakly dissipative equations. Inertial manifolds and normal hyperbolicity. Approximate inertial manifolds of exponential order, Thesis (Ph.D.)–Indiana University, ProQuest LLC, Ann Arbor, MI, 1996.
Informações:
Período letivo: 18 de março a 20 de julho.
Disciplina: MAE511 - Fundamentos de Computação Científica I
Turma: 11356 - Ter/Qui - 15h às 17h
Local: CT, bloco-B, sala 106-B e Google Classroom.
Material: O conteúdo da matéria está disponibilizado online: rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Disciplina com espelho na pós-graduação.
Informações:
Período letivo: 18 de março a 20 de julho.
Disciplina: MAE127 - Equações Diferenciais
Turma: 11096 - Ter/Qui - 10h às 12h
Local: CCMN, F2-021/023 e Google Classroom.
Acesso ao Google Classroom: Todos os alunos devem preencher o Formulário inicial de ED 2024/1 para terem acesso à página da disciplina no Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 19 de março.
Notas de aula: Equações Diferenciais versão 18/mar/2024
Avaliação: Testes regulares assíncronos pelo Google Classroom (semanais ou quinzenais) e provas presenciais.
Informações:
Período letivo: 10 de agosto a 23 de dezembro.
Disciplina: MAE352 - Álgebra Linear Avançada
Turma: 16218 - Ter/Qui - 15h às 17h
Local: D-120 e Google Classroom.
Notas de aula: Álgebra Linear Avançada - versão 4/jan/2024
Avaliação: Listas de exercícios, provas e projeto final.
Informações:
Período letivo: 10 de agosto a 23 de dezembro.
Disciplina: MAE127 - Equações Diferenciais
Turma: 15795 - Ter/Qui - 10h às 12h
Local: CCMN F₂-030 e Google Classroom.
Acesso ao Google Classroom: Todos os alunos devem preencher o Formulário inicial de ED 2023/2 para terem acesso à página da disciplina no Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 11 de agosto.
Notas de aula: Equações Diferenciais versão ago/2022
Avaliação: Testes regulares assíncronos pelo Google Classroom (semanais ou quinzenais) e provas presenciais.
Informações:
Período letivo: 3 de abril a 22 de julho.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/notas_sde e rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Informações:
Período letivo: 3 de abril a 22 de julho.
Disciplina: MAE125 - Álgebra Linear II
Turma: 8039 - EQ (Unificado) - Ter/Qui 8h às 10h
Local: I-119 (CT).
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 4 de abril.
Notas de aula: Livro Curso de Álgebra Linear - 3a. Edição - 2021 - Paulo Goldfeld e Marco A. P. Cabral
Avaliação: Listas pelo Google Classroom e provas presenciais.
Informações:
Período letivo: 29 de agosto a 14 de janeiro.
Disciplina: MAE125 - Álgebra Linear II
Turma: 10821 - POLI/BCMT/IGA (Unificado) - Ter/Qui 8h às 10h
Local: F₂-030 e Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 30 de agosto, na F₂-022.
Notas de aula: Livro Curso de Álgebra Linear - 3a. Edição - 2021 - Paulo Goldfeld e Marco A. P. Cabral
Avaliação: Listas pelo Google Classroom e provas presenciais.
Informações:
Período letivo: 29 de agosto a 14 de janeiro.
Disciplina: MAE127 - Equações Diferenciais
Turma: 8308 - Ter/Qui - 10h às 12h
Local: F₂-022 e Google Classroom.
Acesso ao Google Classroom: Todos os alunos devem preencher o Formulário inicial de ED 2022/2 para terem acesso à página da disciplina no Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 30 de agosto, na F₂-022.
Notas de aula: Equações Diferenciais versão ago/2022
Avaliação: Testes regulares assíncronos pelo Google Classroom (semanais ou quinzenais) e provas presenciais.
Informações:
Período letivo: 11 de abril a 6 de agosto.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/notas_sde e rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Informações:
Período letivo: 11 de abril a 6 de agosto.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/modelagem_matematica/tree/ModMat2022p1 e rmsrosa.github.io/modelagem_matematica. Uma página no Google Classroom também será aberta para a comunição com a turma.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, mini-projetos, apresentações e um projeto final.
Informações:
Período letivo: 16 de novembro a 12 de março.
Local: Google Classroom, Google Meet, YouTube e possivelmente presencialmente, dependendo das condições.
Acesso: Todos os alunos devem preencher o Formulário inicial de Cálculo Infinitesimal II 2021/2 para terem acesso à turma da disciplina no Google Classroom e para serem convidados para os encontros síncronos.
Informações:
Período letivo: 12 de julho a 23 de outubro.
Local: Google Classroom, Google Meet e YouTube.
Acesso: Todos os alunos devem preencher o Formulário inicial de ED 2021/1 para terem acesso à turma da disciplina no Google Classroom e para serem convidados para os encontros síncronos.
Primeira semana: O nosso primeiro encontro será na quinta-feira, dia 15 de julho, através do Google Meet, em link a ser disponibilizado oportunamente.
Carga assíncrona: Vídeo-aulas disponibilizadas semanalmente em canal do YouTube, acessíveis através de links no Google Classroom, assim como a leitura das notas de aula.
Carga síncrona: Uma vez por semana, de 10h às 12h, na terça ou na quinta, via Google Meet, com link disponiblizado pelo Google Classroom ou enviado por convite, com o principal intuito de esclarecer dúvidas surgidas a partir do texto e dos vídeos discutidos na semana.
Notas de aula: Equações Diferenciais versão 2017/1
Avaliação: Atividades semanais pelo Google Classroom e eventuais testes síncronos.
Informações:
Período letivo: 12 de julho a 23 de outubro.
Acesso: Aos alunos que se inscreveram no curso, peço que preencham o Formulário inicial de Modelagem Matemática 2021/1.
Primeira semana: O nosso primeiro encontro será na terça-feira, dia 13 de julho, através do Google Meet, em link a ser disponibilizado oportunamente.
Carga didática: Nossas aulas estão reservadas para as terças e quintas, de 13h às 15h, que serão utilizadas de forma apropriada levando-se em consideração a divisão entre atividades síncronas e assíncronas.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/modelagem_matematica/tree/ModMat2021p1.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, mini-projetos, apresentações e um projeto final.
Informações:
Aos alunos que se inscreveram no curso, peço que preencham o Formulário inicial de Modelagem Matemática 2020/2.
O conteúdo da matéria será disponibilizado em um repositório do github: github.com/rmsrosa/modelagem_matematica/tree/ModMat2020p2.
A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Para a avaliação, devemos ter testes, mini-projetos, apresentações e um projeto final.
Nossas aulas estão reservadas para as terças e quintas, de 13h às 15h, mas ainda falta definir como utilizaremos isso levando-se em consideração a divisão entre atividades síncronas e assíncronas.
Não teremos encontro síncrono na semana de 22 a 26 de março devido à Jornada de Iniciação Científicia, Tecnológica, Artística e Cultural (JICTAC 2020), mas pretendo já começar a divulgar um material inicial nessa semana.
Informações:
PRIMEIRA SEMANA DE AULA: Nesta primeira semana, teremos o nosso encontro pelo Google Meet nesta TERÇA-FEIRA, dia 1/dez, às 10h, pelo link com código woz-tixd-waf.
Local: Google Classroom, Google Meet e YouTube
Acesso: Todos os alunos devem preencher o Formulário inicial de ED 2020/1 para ter acesso à turma da disciplina no Google Classroom e para ser convidado para os encontros síncronos.
Carga assíncrona: Vídeo-aulas disponibilizadas semanalmente em canal do YouTube, acessíveis através de links no Google Classroom, assim como a leitura das notas de aula
Carga síncrona: Uma vez por semana, de 10h às 12h, via Google Meet, com link disponiblizado pelo Google Classroom ou enviado por convite, com o principal intuito de esclarecer dúvidas surgidas a partir do texto e dos vídeos discutidos na semana. O nosso primeiro encontro síncrono será na terça-feira, dia 1/dez/2020, pelo Google Meet. O link para o primeiro encontro será enviado por convite e disponibilizado aqui poucos minutos antes da aula.
Notas de aula: Equações Diferenciais versão 2017/1
Avaliação: Atividades semanais pelo Google Classroom.
Programação:
Início das aulas: 1 de dezembro de 2020
Término das aulas: 6 de março de 2021
Informações:
Local: Google Classroom, Google Meet e YouTube
Acesso: As informações de acesso ao Google Classroom serão enviados por email através do sistema SIGA para os alunos(as) devidamente inscritos(as); mais informações no próprio Google Classroom
Carga assíncrona: Vídeo-aulas disponibilizadas semanalmente em canal do YouTube, acessíveis através de links no Google Classroom, assim como a leitura das notas de aula
Carga síncrona: Uma vez por semana, de 10h às 12h, via Google Meet, com link disponiblizado pelo Google Classroom, com o principal intuito de esclarecer dúvidas surgidas a partir do texto e dos vídeos discutidos na semana
Notas de aula: Equações Diferenciais versão 2017/1
Avaliação: Atividades semanais pelo Google Classroom.
Programação:
Início das aulas: 26 de agosto de 2020
Término das aulas: 19 de novembro de 2020
Informações:
Continuação de projeto extracurricular iniciado em março de 2020
Informações:
Sala de aula: F_2 026 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 5 de agosto de 2019
Informações:
Sala de aula: F_2 026 (CCMN)
Dias das aulas: Segundas e sextas, das 13h às 15h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo III Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 5 de agosto de 2018
Informações:
Aulas: Terças e quintas de 13h às 15h
Salas de aula: LIG ABC-119 (CT) às terças e A204 (CT) às quintas
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 11 de março de 2019
Material:
Informações:
Sala de aula: B-110 (CT)
Aulas: Terças e quintas de 15h às 17h
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 11 de março de 2019
Primeira Prova: 9 de maio de 2019 (quinta-feira)
Segunda Prova: 25 de junho de 2019 (terça-feira)
Prova Final: 4 de julho de 2019 (quinta-feira)
Material:
Informações:
Sala de aula: F_2 010 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 6 de agosto de 2018
Informações:
Sala de aula: F_2 010 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 12 de março de 2018
Informações:
Sala de aula: F_2 010 (CCMN)
Dias das aulas: Terças e quintas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Álgebra Linear II Unificada
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 13 de março de 2018
Informações:
Sala de aula: F_2 014 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 31 de julho de 2017
Informações:
Sala de aula: F_2 010 (CCMN)
Dias e horários das aulas: Quartas e sextas, das 13h às 15h
Monitor: Lucas Moura, sala ABC-116
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 2 de agosto de 2017 (quarta-feira)
Primeira Prova: 29 de setembro de 2017 (sexta-feira)
Segunda Prova: 29 de novembro de 2017 (quarta-feira)
Prova Final: 8 de dezembro de 2017 (sexta-feira)
Material:
Primeira Lista - entregar na aula do dia 25 de agosto, sexta-feira
Segunda Lista - entregar na aula do dia 27 de agosto, quarta-feira (ou enviar pdf por email até o final do horário da aula)
Terceira Lista - entregar na aula do dia 18 de outubro, quarta-feira (ou enviar pdf por email até as 15h desse dia)
Quarta Lista - entregar na aula do dia 8 de novembro, quarta-feira (ou enviar pdf por email até as 15h desse dia)
Quinta Lista - não precisa entregar; não será corrigida
Informações:
Sala de aula: B-110 (CT)
Aulas: Terças e quintas de 10h às 12h
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 7 de março de 2017
Primeira Prova: 20 de abril de 2017 (quinta-feira)
Segunda Prova: 22 de junho de 2017 (quinta-feira)
Prova Final: 29 de junho de 2017 (quinta-feira)
Material:
Informações:
Sala de aula: F_2 010 (CCMN)
Aulas: segundas e quartas, de 10h às 12h
Unificado com a Engenharia. Mais informações sobre o curso unificado na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Informações:
Sala de aula: F_3 014 (CCMN)
Aulas: quartas e sextas, de 13h às 15h
Cálculo da Média: M = (P1/3 + 2 T1/3)/2 + P2/2
Cálculo da Média Final: MF = (M+PF)/2
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 31 de agosto de 2016
Primeira Prova: 4 de novembro de 2016 (sexta-feira)
Trabalho: entregar as soluções da P1 na aula do dia 11/nov/2016
Segunda Prova: 14 de dezembro de 2016 (quarta-feira)
Prova Final: 21 de dezembro de 2016 (quarta-feira)
Material:
Material: Modelagem Matemática (notas de aula)
Material: Modelagem Matemática (notas de aula)
Material: Modelagem Matemática (notas de aula)
Informações:
Período letivo: 18 de março a 20 de julho.
Disciplina: MAE731 - Tópicos de Matemática Aplicada I
Turma: 5058 - Ter/Qui - 15h às 17h
Local: CT, bloco-B, sala 106-B e Google Classroom.
Material: O conteúdo da matéria está disponibilizado online: rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Disciplina com espelho na graduação.
Informações:
Período letivo: 10 de agosto a 23 de dezembro.
Local: D-120 e Google Classroom.
Disciplina: MAE709 - Álgebra Linear (Mestrado)
Turma: 2189 - Ter/Qui - 15h às 17h
Notas de aula: Álgebra Linear Avançada - versão 4/jan/2024
Avaliação: Listas de exercícios, provas e projeto final.
Informações:
Período letivo: 3 de abril a 22 de julho.
Aula com espelho na graduação. Mais informações acima.
Informações:
Sala de aula: B-106b (CT)
Dias e horários das aulas: terças e quintas, das 13h às 15h
Meu gabinete: C-113b (CT)
Programação:
Início das aulas: 7 de agosto de 2018 (terça-feira)
Primeira Prova: 30 de outubro de 2018, de 12h30 às 15h (terça-feira)
Segunda Prova: 8 de novembro de 2018, de 12h00 às 15h (quinta-feira)
Terceira Prova: 11 de dezembro de 2018 (terça-feira)
Material:
Evento: Primeira Escola Brasileira de Equações Diferenciais (I EBED)
Local: IMECC-Unicamp, Campinas, SP
Datas: de 9 a 13 de junho de 2003
Material: Transparências utilizadas
Evento: Programa de Verão do LNCC
Local: LNCC, Petrópolis, RJ
Datas: de 24 a 27 de fevereiro de 2003
Material: Transparências utilizadas
Evento: Programa de Verão do IM-UFRJ
Local: Instituto de Matemática da UFRJ
Datas: 4, 6, 8, 18, 20 e 22 de fevereiro de 2002
Evento: Seventh Paseky School on Mathematical Theory in Fluid Mechanics
Local: Paseky, República Tcheca
Datas: 3 a 10 de junho de 2001
Evento: Minicurso no IM-UFRJ
Local: Instituto de Matemática da UFRJ
Datas: julho de 1995 (durante o meu doutorado, em uma visita ao Rio)
Anne Caroline Bronzi, "Soluções estatísticas das equações de Navier-Stokes", ago/2010 a jul/2012.
Cecília Freire Mondaini, "Soluções estatísticas das equações de Navier-Stokes", 20 de maio de 2014.
Fábio Antônio Tavares Ramos, "Estimativas estatísticas e dissipação anômola em turbulência de fluidos", 3 de agosto de 2007.
Cecília Freire Mondaini, "Uma formulação abstrata para o estudo de soluções estatísticas das equações de Navier-Stokes", 30 de março de 2010.
Raphael Carlos Santos Machado, "Criptossistemas dinâmicos", 31 de julho de 2006.
Adriano Maurício de Almeida Côrtes, "Conjuntos de Aubry-Mather no contexto dos mapeamentos twist", 31 de maio de 2006.
Fábio Antônio Tavares Ramos, "Métodos geométricos em transporte lagrangiano", 22 de agosto de 2003.
Luciana Santos da Silva, "Solução Numérica de um Escoamento Geofísico Aplicado ao Rio Amazonas", 20 de dezembro de 2002.
Sérgio Krakowski, "Estudo de caos em um sistema dinâmico de dimensão infinita", 27 de maio de 2002.
Patrícia Sanez, "Atrator global para as equações de Navier-Stokes em domínios irregulares", 30 de março de 2001.
Iniciações Científicas em 2010
Felipe de Souza Valladão "Equações diferenciais"
Renan Reimermendt "Equações diferenciais"
Iniciações Científicas em 2007
Cristina Pimenta de Mello Spineti, "Equações diferenciais e biomatemática"
Iniciações Científicas em 2006
Nikolas Lippmann Pareschi, "Equações diferenciais e biomatemática"
Cristina Pimenta de Mello Spineti, "Equações diferenciais e biomatemática"
Iniciações Científicas em 2005
Nikolas Lippmann Pareschi, "Equações diferenciais e biomatemática"
Cristina Pimenta de Mello Spineti, "Equações diferenciais e biomatemática"
Ronald Simões de Mattos Pinto, "Equações diferenciais e mecânica celeste"
German Lourenço Mejia, "Equações diferenciais"
Trabalhos de Conclusão de Curso em 2005
Carlos Ferreira, "Sistemas Hamiltonianos e aplicações em Modelagem Molecular"
Iniciações Científicas em 2004
Rogério Lourenço Fernandez, "Sistemas Dinâmicos e Bifurcações"
Glauber Ferreira de Medeiros, "Sistemas Dinâmicos e aplicações no estudo de Quorum Sensing em uma colônia de bactérias"
Ronaldo Alves de Abreu, "Sistemas Dinâmicos"
Trabalhos de Conclusão de Curso em 2004:
Carlos Ferreira, "Sistemas Hamiltonianos e aplicações em Modelagem Molecular"
Iniciações Científicas em 2003:
Rogério Lourenço Fernandez, "Sistemas Dinâmicos e Bifurcações"
Rafael Brandão de Rezende Borges, "EDP e Análise Numérica"
Glauber Ferreira de Medeiros, "Sistemas Dinâmicos e aplicações no estudo de Quorum Sensing em uma colônia de bactérias"
Leandro de Souza Gonçalves, "Sistemas Dinâmicos e aplicações no estudo de Quorum Sensing em uma colônia de bactérias"
Trabalhos de Conclusão de Curso em 2003:
Carlos Ferreira, "Sistemas Hamiltonianos e aplicações em Modelagem Molecular"
Informações:
Período letivo: 12 de agosto a 14 de dezembro.
Disciplina: MAE352 - Álgebra Linear Avançada (com espelho na pós-graduação MAE709 - Álgebra Linear)
Turma: 0000 - Ter/Qui - 10h às 12h
Local: B-106A e Google Classroom.
Notas de aula: Álgebra Linear Avançada - versão 4/jan/2024
Avaliação: Listas de exercícios, provas e projeto final.
Informações:
Período letivo: 18 de março a 20 de julho.
Disciplina: MAE511 - Fundamentos de Computação Científica I
Turma: 11356 - Ter/Qui - 15h às 17h
Local: CT, bloco-B, sala 106-B e Google Classroom.
Material: O conteúdo da matéria está disponibilizado online: rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Disciplina com espelho na pós-graduação.
Informações:
Período letivo: 18 de março a 20 de julho.
Disciplina: MAE127 - Equações Diferenciais
Turma: 11096 - Ter/Qui - 10h às 12h
Local: CCMN, F2-021/023 e Google Classroom.
Acesso ao Google Classroom: Todos os alunos devem preencher o Formulário inicial de ED 2024/1 para terem acesso à página da disciplina no Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 19 de março.
Notas de aula: Equações Diferenciais versão 18/mar/2024
Avaliação: Testes regulares assíncronos pelo Google Classroom (semanais ou quinzenais) e provas presenciais.
Informações:
Período letivo: 10 de agosto a 23 de dezembro.
Disciplina: MAE352 - Álgebra Linear Avançada
Turma: 16218 - Ter/Qui - 15h às 17h
Local: D-120 e Google Classroom.
Notas de aula: Álgebra Linear Avançada - versão 4/jan/2024
Avaliação: Listas de exercícios, provas e projeto final.
Informações:
Período letivo: 10 de agosto a 23 de dezembro.
Disciplina: MAE127 - Equações Diferenciais
Turma: 15795 - Ter/Qui - 10h às 12h
Local: CCMN F₂-030 e Google Classroom.
Acesso ao Google Classroom: Todos os alunos devem preencher o Formulário inicial de ED 2023/2 para terem acesso à página da disciplina no Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 11 de agosto.
Notas de aula: Equações Diferenciais versão ago/2022
Avaliação: Testes regulares assíncronos pelo Google Classroom (semanais ou quinzenais) e provas presenciais.
Informações:
Período letivo: 3 de abril a 22 de julho.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/notas_sde e rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Informações:
Período letivo: 3 de abril a 22 de julho.
Disciplina: MAE125 - Álgebra Linear II
Turma: 8039 - EQ (Unificado) - Ter/Qui 8h às 10h
Local: I-119 (CT).
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 4 de abril.
Notas de aula: Livro Curso de Álgebra Linear - 3a. Edição - 2021 - Paulo Goldfeld e Marco A. P. Cabral
Avaliação: Listas pelo Google Classroom e provas presenciais.
Informações:
Período letivo: 29 de agosto a 14 de janeiro.
Disciplina: MAE125 - Álgebra Linear II
Turma: 10821 - POLI/BCMT/IGA (Unificado) - Ter/Qui 8h às 10h
Local: F₂-030 e Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 30 de agosto, na F₂-022.
Notas de aula: Livro Curso de Álgebra Linear - 3a. Edição - 2021 - Paulo Goldfeld e Marco A. P. Cabral
Avaliação: Listas pelo Google Classroom e provas presenciais.
Informações:
Período letivo: 29 de agosto a 14 de janeiro.
Disciplina: MAE127 - Equações Diferenciais
Turma: 8308 - Ter/Qui - 10h às 12h
Local: F₂-022 e Google Classroom.
Acesso ao Google Classroom: Todos os alunos devem preencher o Formulário inicial de ED 2022/2 para terem acesso à página da disciplina no Google Classroom.
Primeira aula: O nosso primeiro encontro será na terça-feira, dia 30 de agosto, na F₂-022.
Notas de aula: Equações Diferenciais versão ago/2022
Avaliação: Testes regulares assíncronos pelo Google Classroom (semanais ou quinzenais) e provas presenciais.
Informações:
Período letivo: 11 de abril a 6 de agosto.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/notas_sde e rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Informações:
Período letivo: 11 de abril a 6 de agosto.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/modelagem_matematica/tree/ModMat2022p1 e rmsrosa.github.io/modelagem_matematica. Uma página no Google Classroom também será aberta para a comunição com a turma.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, mini-projetos, apresentações e um projeto final.
Informações:
Período letivo: 16 de novembro a 12 de março.
Local: Google Classroom, Google Meet, YouTube e possivelmente presencialmente, dependendo das condições.
Acesso: Todos os alunos devem preencher o Formulário inicial de Cálculo Infinitesimal II 2021/2 para terem acesso à turma da disciplina no Google Classroom e para serem convidados para os encontros síncronos.
Informações:
Período letivo: 12 de julho a 23 de outubro.
Local: Google Classroom, Google Meet e YouTube.
Acesso: Todos os alunos devem preencher o Formulário inicial de ED 2021/1 para terem acesso à turma da disciplina no Google Classroom e para serem convidados para os encontros síncronos.
Primeira semana: O nosso primeiro encontro será na quinta-feira, dia 15 de julho, através do Google Meet, em link a ser disponibilizado oportunamente.
Carga assíncrona: Vídeo-aulas disponibilizadas semanalmente em canal do YouTube, acessíveis através de links no Google Classroom, assim como a leitura das notas de aula.
Carga síncrona: Uma vez por semana, de 10h às 12h, na terça ou na quinta, via Google Meet, com link disponiblizado pelo Google Classroom ou enviado por convite, com o principal intuito de esclarecer dúvidas surgidas a partir do texto e dos vídeos discutidos na semana.
Notas de aula: Equações Diferenciais versão 2017/1
Avaliação: Atividades semanais pelo Google Classroom e eventuais testes síncronos.
Informações:
Período letivo: 12 de julho a 23 de outubro.
Acesso: Aos alunos que se inscreveram no curso, peço que preencham o Formulário inicial de Modelagem Matemática 2021/1.
Primeira semana: O nosso primeiro encontro será na terça-feira, dia 13 de julho, através do Google Meet, em link a ser disponibilizado oportunamente.
Carga didática: Nossas aulas estão reservadas para as terças e quintas, de 13h às 15h, que serão utilizadas de forma apropriada levando-se em consideração a divisão entre atividades síncronas e assíncronas.
Material: O conteúdo da matéria será disponibilizado, ao longo do curso, em um repositório do github: github.com/rmsrosa/modelagem_matematica/tree/ModMat2021p1.
Ferramentas computacionais: A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Avaliação: Para a avaliação, devemos ter testes, mini-projetos, apresentações e um projeto final.
Informações:
Aos alunos que se inscreveram no curso, peço que preencham o Formulário inicial de Modelagem Matemática 2020/2.
O conteúdo da matéria será disponibilizado em um repositório do github: github.com/rmsrosa/modelagem_matematica/tree/ModMat2020p2.
A parte computacional será desenvolvida na linguagem de programação Julia e poderá ser feita remotamente (na nuvem, em servidores no binder) ou localmente, em suas máquinas.
Para a avaliação, devemos ter testes, mini-projetos, apresentações e um projeto final.
Nossas aulas estão reservadas para as terças e quintas, de 13h às 15h, mas ainda falta definir como utilizaremos isso levando-se em consideração a divisão entre atividades síncronas e assíncronas.
Não teremos encontro síncrono na semana de 22 a 26 de março devido à Jornada de Iniciação Científicia, Tecnológica, Artística e Cultural (JICTAC 2020), mas pretendo já começar a divulgar um material inicial nessa semana.
Informações:
PRIMEIRA SEMANA DE AULA: Nesta primeira semana, teremos o nosso encontro pelo Google Meet nesta TERÇA-FEIRA, dia 1/dez, às 10h, pelo link com código woz-tixd-waf.
Local: Google Classroom, Google Meet e YouTube
Acesso: Todos os alunos devem preencher o Formulário inicial de ED 2020/1 para ter acesso à turma da disciplina no Google Classroom e para ser convidado para os encontros síncronos.
Carga assíncrona: Vídeo-aulas disponibilizadas semanalmente em canal do YouTube, acessíveis através de links no Google Classroom, assim como a leitura das notas de aula
Carga síncrona: Uma vez por semana, de 10h às 12h, via Google Meet, com link disponiblizado pelo Google Classroom ou enviado por convite, com o principal intuito de esclarecer dúvidas surgidas a partir do texto e dos vídeos discutidos na semana. O nosso primeiro encontro síncrono será na terça-feira, dia 1/dez/2020, pelo Google Meet. O link para o primeiro encontro será enviado por convite e disponibilizado aqui poucos minutos antes da aula.
Notas de aula: Equações Diferenciais versão 2017/1
Avaliação: Atividades semanais pelo Google Classroom.
Programação:
Início das aulas: 1 de dezembro de 2020
Término das aulas: 6 de março de 2021
Informações:
Local: Google Classroom, Google Meet e YouTube
Acesso: As informações de acesso ao Google Classroom serão enviados por email através do sistema SIGA para os alunos(as) devidamente inscritos(as); mais informações no próprio Google Classroom
Carga assíncrona: Vídeo-aulas disponibilizadas semanalmente em canal do YouTube, acessíveis através de links no Google Classroom, assim como a leitura das notas de aula
Carga síncrona: Uma vez por semana, de 10h às 12h, via Google Meet, com link disponiblizado pelo Google Classroom, com o principal intuito de esclarecer dúvidas surgidas a partir do texto e dos vídeos discutidos na semana
Notas de aula: Equações Diferenciais versão 2017/1
Avaliação: Atividades semanais pelo Google Classroom.
Programação:
Início das aulas: 26 de agosto de 2020
Término das aulas: 19 de novembro de 2020
Informações:
Continuação de projeto extracurricular iniciado em março de 2020
Informações:
Sala de aula: F_2 026 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 5 de agosto de 2019
Informações:
Sala de aula: F_2 026 (CCMN)
Dias das aulas: Segundas e sextas, das 13h às 15h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo III Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 5 de agosto de 2018
Informações:
Aulas: Terças e quintas de 13h às 15h
Salas de aula: LIG ABC-119 (CT) às terças e A204 (CT) às quintas
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 11 de março de 2019
Material:
Informações:
Sala de aula: B-110 (CT)
Aulas: Terças e quintas de 15h às 17h
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 11 de março de 2019
Primeira Prova: 9 de maio de 2019 (quinta-feira)
Segunda Prova: 25 de junho de 2019 (terça-feira)
Prova Final: 4 de julho de 2019 (quinta-feira)
Material:
Informações:
Sala de aula: F_2 010 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 6 de agosto de 2018
Informações:
Sala de aula: F_2 010 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 12 de março de 2018
Informações:
Sala de aula: F_2 010 (CCMN)
Dias das aulas: Terças e quintas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Álgebra Linear II Unificada
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 13 de março de 2018
Informações:
Sala de aula: F_2 014 (CCMN)
Dias das aulas: Segundas e quartas, das 10h às 12h
Turma de curso unificado: Mais informações sobre o curso na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 31 de julho de 2017
Informações:
Sala de aula: F_2 010 (CCMN)
Dias e horários das aulas: Quartas e sextas, das 13h às 15h
Monitor: Lucas Moura, sala ABC-116
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 2 de agosto de 2017 (quarta-feira)
Primeira Prova: 29 de setembro de 2017 (sexta-feira)
Segunda Prova: 29 de novembro de 2017 (quarta-feira)
Prova Final: 8 de dezembro de 2017 (sexta-feira)
Material:
Primeira Lista - entregar na aula do dia 25 de agosto, sexta-feira
Segunda Lista - entregar na aula do dia 27 de agosto, quarta-feira (ou enviar pdf por email até o final do horário da aula)
Terceira Lista - entregar na aula do dia 18 de outubro, quarta-feira (ou enviar pdf por email até as 15h desse dia)
Quarta Lista - entregar na aula do dia 8 de novembro, quarta-feira (ou enviar pdf por email até as 15h desse dia)
Quinta Lista - não precisa entregar; não será corrigida
Informações:
Sala de aula: B-110 (CT)
Aulas: Terças e quintas de 10h às 12h
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 7 de março de 2017
Primeira Prova: 20 de abril de 2017 (quinta-feira)
Segunda Prova: 22 de junho de 2017 (quinta-feira)
Prova Final: 29 de junho de 2017 (quinta-feira)
Material:
Informações:
Sala de aula: F_2 010 (CCMN)
Aulas: segundas e quartas, de 10h às 12h
Unificado com a Engenharia. Mais informações sobre o curso unificado na página Cálculo II Unificado
Meu gabinete: C-113B (CT)
Informações:
Sala de aula: F_3 014 (CCMN)
Aulas: quartas e sextas, de 13h às 15h
Cálculo da Média: M = (P1/3 + 2 T1/3)/2 + P2/2
Cálculo da Média Final: MF = (M+PF)/2
Meu gabinete: C-113B (CT)
Programação:
Início das aulas: 31 de agosto de 2016
Primeira Prova: 4 de novembro de 2016 (sexta-feira)
Trabalho: entregar as soluções da P1 na aula do dia 11/nov/2016
Segunda Prova: 14 de dezembro de 2016 (quarta-feira)
Prova Final: 21 de dezembro de 2016 (quarta-feira)
Material:
Material: Modelagem Matemática (notas de aula)
Material: Modelagem Matemática (notas de aula)
Material: Modelagem Matemática (notas de aula)
Informações:
Período letivo: 12 de agosto a 14 de dezembro
Disciplina: MAE709 - Álgebra Linear (com espelho na graduação MAE352 - Álgebra Linear Avançada)
Turma: 0000 - Ter/Qui - 10h às 12h
Local: B-106A e Google Classroom.
Notas de aula: Álgebra Linear Avançada - versão 4/jan/2024
Avaliação: Listas de exercícios, provas e projeto final.
Informações:
Período letivo: 12 de agosto a 14 de dezembro
Disciplina: MAE711 - Tópicos de Matemática Aplicada I
Turma: 0000 - Seg/Qua - 10h às 12h
Local: B-108B e Google Classroom
Material: O conteúdo da matéria está disponibilizado online: https://rmsrosa.github.io/random_notes/dev/generative/overview/.
Avaliação: Para a avaliação, devemos ter projetos e apresentações.
Informações:
Período letivo: 18 de março a 20 de julho.
Disciplina: MAE721 - Tópicos de Matemática Aplicada II
Turma: 5058 - Ter/Qui - 15h às 17h
Local: CT, bloco-B, sala 106-B e Google Classroom.
Material: O conteúdo da matéria está disponibilizado online: rmsrosa.github.io/notas_sde. Uma página no Google Classroom também será aberta para a comunição com a turma.
Avaliação: Para a avaliação, devemos ter testes, projetos e apresentações.
Disciplina com espelho na graduação.
Informações:
Período letivo: 10 de agosto a 23 de dezembro.
Local: D-120 e Google Classroom.
Disciplina: MAE709 - Álgebra Linear (Mestrado)
Turma: 2189 - Ter/Qui - 15h às 17h
Notas de aula: Álgebra Linear Avançada - versão 4/jan/2024
Avaliação: Listas de exercícios, provas e projeto final.
Informações:
Período letivo: 3 de abril a 22 de julho.
Aula com espelho na graduação. Mais informações acima.
Informações:
Sala de aula: B-106b (CT)
Dias e horários das aulas: terças e quintas, das 13h às 15h
Meu gabinete: C-113b (CT)
Programação:
Início das aulas: 7 de agosto de 2018 (terça-feira)
Primeira Prova: 30 de outubro de 2018, de 12h30 às 15h (terça-feira)
Segunda Prova: 8 de novembro de 2018, de 12h00 às 15h (quinta-feira)
Terceira Prova: 11 de dezembro de 2018 (terça-feira)
Material:
Evento: Primeira Escola Brasileira de Equações Diferenciais (I EBED)
Local: IMECC-Unicamp, Campinas, SP
Datas: de 9 a 13 de junho de 2003
Material: Transparências utilizadas
Evento: Programa de Verão do LNCC
Local: LNCC, Petrópolis, RJ
Datas: de 24 a 27 de fevereiro de 2003
Material: Transparências utilizadas
Evento: Programa de Verão do IM-UFRJ
Local: Instituto de Matemática da UFRJ
Datas: 4, 6, 8, 18, 20 e 22 de fevereiro de 2002
Evento: Seventh Paseky School on Mathematical Theory in Fluid Mechanics
Local: Paseky, República Tcheca
Datas: 3 a 10 de junho de 2001
Evento: Minicurso no IM-UFRJ
Local: Instituto de Matemática da UFRJ
Datas: julho de 1995 (durante o meu doutorado, em uma visita ao Rio)
Anne Caroline Bronzi, "Soluções estatísticas das equações de Navier-Stokes", ago/2010 a jul/2012.
Cecília Freire Mondaini, "Soluções estatísticas das equações de Navier-Stokes", 20 de maio de 2014.
Fábio Antônio Tavares Ramos, "Estimativas estatísticas e dissipação anômola em turbulência de fluidos", 3 de agosto de 2007.
Cecília Freire Mondaini, "Uma formulação abstrata para o estudo de soluções estatísticas das equações de Navier-Stokes", 30 de março de 2010.
Raphael Carlos Santos Machado, "Criptossistemas dinâmicos", 31 de julho de 2006.
Adriano Maurício de Almeida Côrtes, "Conjuntos de Aubry-Mather no contexto dos mapeamentos twist", 31 de maio de 2006.
Fábio Antônio Tavares Ramos, "Métodos geométricos em transporte lagrangiano", 22 de agosto de 2003.
Luciana Santos da Silva, "Solução Numérica de um Escoamento Geofísico Aplicado ao Rio Amazonas", 20 de dezembro de 2002.
Sérgio Krakowski, "Estudo de caos em um sistema dinâmico de dimensão infinita", 27 de maio de 2002.
Patrícia Sanez, "Atrator global para as equações de Navier-Stokes em domínios irregulares", 30 de março de 2001.
Iniciações Científicas em 2010
Felipe de Souza Valladão "Equações diferenciais"
Renan Reimermendt "Equações diferenciais"
Iniciações Científicas em 2007
Cristina Pimenta de Mello Spineti, "Equações diferenciais e biomatemática"
Iniciações Científicas em 2006
Nikolas Lippmann Pareschi, "Equações diferenciais e biomatemática"
Cristina Pimenta de Mello Spineti, "Equações diferenciais e biomatemática"
Iniciações Científicas em 2005
Nikolas Lippmann Pareschi, "Equações diferenciais e biomatemática"
Cristina Pimenta de Mello Spineti, "Equações diferenciais e biomatemática"
Ronald Simões de Mattos Pinto, "Equações diferenciais e mecânica celeste"
German Lourenço Mejia, "Equações diferenciais"
Trabalhos de Conclusão de Curso em 2005
Carlos Ferreira, "Sistemas Hamiltonianos e aplicações em Modelagem Molecular"
Iniciações Científicas em 2004
Rogério Lourenço Fernandez, "Sistemas Dinâmicos e Bifurcações"
Glauber Ferreira de Medeiros, "Sistemas Dinâmicos e aplicações no estudo de Quorum Sensing em uma colônia de bactérias"
Ronaldo Alves de Abreu, "Sistemas Dinâmicos"
Trabalhos de Conclusão de Curso em 2004:
Carlos Ferreira, "Sistemas Hamiltonianos e aplicações em Modelagem Molecular"
Iniciações Científicas em 2003:
Rogério Lourenço Fernandez, "Sistemas Dinâmicos e Bifurcações"
Rafael Brandão de Rezende Borges, "EDP e Análise Numérica"
Glauber Ferreira de Medeiros, "Sistemas Dinâmicos e aplicações no estudo de Quorum Sensing em uma colônia de bactérias"
Leandro de Souza Gonçalves, "Sistemas Dinâmicos e aplicações no estudo de Quorum Sensing em uma colônia de bactérias"
Trabalhos de Conclusão de Curso em 2003:
Carlos Ferreira, "Sistemas Hamiltonianos e aplicações em Modelagem Molecular"
Level sets of vorticity in a periodic, two-mode-forced incompressible 2D Navier-Stokes pseudo-spectrally simulated flow. (Click here for an extended simulation))
Turbulence is a fascinating phenomenom. It occurs in many types of fluids under various conditions. From the most visible ones, as in the formation of currents in rivers, to more hidden ones, as the flow of blood and other fluids in our bodies, of oil and gas in pipelines and wellbores, and of magma in the Earth's mantle and in magma chambers.
Despite the fact that there are well-accepted mathematical models to describe the motion of the associated fluid flows, explicit solutions to the modeling systems of equations exist only in very particular cases, so that alternative routes are needed.
Numerical solutions are largely exploited in engineering problems. However, in the case of turbulent flows, the complexity of the flow is such that the required mesh resolution for numerically modelling the dynamics of all the energetic spatial and temporal structures of the flow is impracticle for the current and even forthcoming computational power. Hence, appropriate simplifications and lower dimensional models must still be used. For this reason, research in turbulence is still a very active field.
My research mainly focuses on the system of equations known as the homogeneous and incompressible Navier-Stokes equations (iNSE), which is a model for the flow of certain Newtonian fluids such as water and oil, under most conditions. In a sense, this is the simplest possible type of fluid, but which already presents many theoretical and practical challenges.
In particular, the question of well-posedness of the iNSE is still an open problem and is currently a point of intense research due to some recent advances in similar problems. This question is one of the Millenium Problems, for which there is a one-million dolar prize.
My interest, however, is mostly devoted to building a mathematical foundation for rigorous results related to turbulence. This comprises
looking for estimates of turbulence-related time-averaged mean quantities for Leray-Hopf weak solutions of the iNSE; developing and analysing a proper framework for statistical solutions of the iNSE as a rigorous way of assessing ensemble averages of the flow; and numerically investigating the results associated with the previous two items.
Representation of a push-forward of an initial measure by a semigroup (top); a phase-space statistical solution (middle); and a trajectory statistical solution (bottom).
This work stemmed from the study of statistical solutions of the Navier-Stokes. It aims towards a generalization of this notion to various types of evolution equations.
An abstract framework is formulated in which the phase-space of the evolutionary system is assumed to be a Hausdorff topological space, and the evolution equation is considered in a weak sense, in the dual of a topological vector space.
The main results obtained so far in this context are
the existence of solutions to the associated initial value problem under simple and natural conditions;
a vast number of applications showing the applicability of the theory; and
conditions for the convergence of subnets of statistical solutions depending on parameters of a family of equations.
There are challenging functional-analytic and measure-theoretic problems involved in this line of research due to the high degree of abstraction and the need to be as applicable as possible.
The "Dynamic Sugar Loaf", made of a heteroclinic cycle, a repeller focus as the Sun, and water waves completing the picture of a famous landmark in Rio de Janeiro.
Several real-life phenomena and advanced technological problems are modeled with systems involving evolutionary partial differential equations. Such systems appear in a multitude of areas, such as Physics, Engineering, Biology, Chemistry, Biochemistry, Finance, and so on.
As evolutionary partial differential equations, they may generate dynamical systems in phase spaces which are of infinite dimension. Many of these systems have very rich dynamics.
The study of evolutionary partial differential equations is thus of great interest and very challenging.
I started my research precisely in this area, working on it since my MSc at IM-UFRJ, continuing on it during my PhD at the Indiana University and thereafter. The study of evolutionary nonlinear partial differential equations as infinite-dimensional dynamical system was just taking off prior to my MSc and have been flourishing since then.
I have worked on several different types of nonlinear partial differential equations, from reaction-diffusion equations, to fluid-flow, dispersive, and wave-type equations. Currently, I have been working on fluid-flow models with viscous heating, important in some situations of magma flows and certain oil flows in pipelines.
Important questions in this field include
assessing the local well posedness of the system;
assessing the global well-posedness of the system and the existence of an associated dynamical system;
understanding the complexity of the associated dynamics;
analysing the possible finite-dimensionality of the asymptotic behavior of the system;
looking for finite-dimensional systems mimicking or approximating the dynamics of the system;
understanding the effect of the approximation of the infinite-dimensional system by finite-dimensional numerical schemes;
exploiting the finite dimensional asymptotic behavior of some system for control purposes; and so on.
Deep water prospection aspects
The salt layer, below, in yellow, modeled as a visco-elastic fluid, and being deformed into "mushroom"-like diapirs (geological intrusion), under the load of different layers of elastic sedimentary rocks.
Pre-salt oil became a fundamental asset for Brazil, but its exploitation is an extraordinary task.
The oil prospected before the pre-salt era lies in sedimentary layers at the bottom of the ocean or below ground. The pre-salt oil, on the other hand, lies within a carbonaceous layer, just below a thick 2km salt layer, which is below the sedimentary layers, and at the bottom of the ocean. The difficulties for prospection are enormous.
The aim of this project was initially to find a better model for the kinematics of the salt layer and then to study the behavior of the salt layer according to this model.
Salt rocks in the Earth's crust behave as solids on small time scales, but display a viscoelastic behavior on intermediate and long time scales. Some works assume a Newtonian viscous model or an Oldroyd-B visco-elastic model. We consider, on the other hand, a more general visco-elastic model proposed by I-Shi Liu (IM-UFRJ) that we termed the Mooney-Rivlin-Liu model.
Using experimental data obtained from a number of tests on samples of different types of salt rocks, we have showed that our model fits considerably better the observed deformations.
Having validated the model, we studied its analytical properties as well as some qualitive properties relevant for
the evolution of the salt layer in the Earth's crust;
the formation of salt diapirs flowing into the upper sediment layer;
the mechanical properties involved in the drilling process of boreholes through the salt layer; and
the optimization of the drilling path throught the different layers.
This is a joint project with CENPES/PETROBRÁS. We are now renewing this project to continue researching the properties of the salt layer and to include developing a model for the pre-salt layer and analysing the interaction between all those layers according to these models.
These diverse studies involve analysis of partial differential equations, dynamical systems, optimization methods, numerical analysis, and computer simulations.
Hop flowers and a plot of an experimental data of the humulone to iso-humulone conversion and a fitting by a saturation model.
There is a multitude of physical, chemical, and biological processes ocurring during the brewing and storage stages of beer manufacture.
Many of these processes can be given a mathematical formulation, and most of them can be modeled with systems of ordinary or partial differential equations. These are the sort of models that interest me the most (besides drinking the end product, of course).
From the physicochemical processes that turn the starch within the barley malt (one of the four main ingredients of beer) into the sugary wort that feeds the yeast (the most fundamental ingredient of beer); to the physicochemical process that turn the bitter and slightly soluble alpha-acids present in the hops (the third main ingredient of beer in this list) into the much more bitter and more soluble iso-alpha-acids that help balance the lingering sweetness that remains from the wort; to the biochemical processes involved when the yeast turns most of the sugar in the wort into alchohol and other aromatic compounds that make for a delicious (or not) beer. These are typical processes where ordinary differential equations can be used as models. (In case you notice anything missing, the fourth ingredient not mentioned yet is the most abundant one, namely water, which serves as the solvent for the other compounds.)
In most situations these processes can be assumed to occur within a spatially homogeneous and static medium, leading to systems of ordinary differential equations. Sometimes, however, the proccess requires the media to be treated as a flowing fluid, either as Newtonian-type flow inside a mash tun, a kettle, or a fermentation tank, or as a fluid flowing through a porous bed of spent grains, as in the sparging step. These processes lead to partial differential equations and are particularly important in the very design of the equipment, for efficiency (optimization) purposes.
Level sets of vorticity in a periodic, two-mode-forced incompressible 2D Navier-Stokes pseudo-spectrally simulated flow. (Click here for an extended simulation))
Turbulence is a fascinating phenomenom. It occurs in many types of fluids under various conditions. From the most visible ones, as in the formation of currents in rivers, to more hidden ones, as the flow of blood and other fluids in our bodies, of oil and gas in pipelines and wellbores, and of magma in the Earth's mantle and in magma chambers.
Despite the fact that there are well-accepted mathematical models to describe the motion of the associated fluid flows, explicit solutions to the modeling systems of equations exist only in very particular cases, so that alternative routes are needed.
Numerical solutions are largely exploited in engineering problems. However, in the case of turbulent flows, the complexity of the flow is such that the required mesh resolution for numerically modelling the dynamics of all the energetic spatial and temporal structures of the flow is impracticle for the current and even forthcoming computational power. Hence, appropriate simplifications and lower dimensional models must still be used. For this reason, research in turbulence is still a very active field.
My research mainly focuses on the system of equations known as the homogeneous and incompressible Navier-Stokes equations (iNSE), which is a model for the flow of certain Newtonian fluids such as water and oil, under most conditions. In a sense, this is the simplest possible type of fluid, but which already presents many theoretical and practical challenges.
In particular, the question of well-posedness of the iNSE is still an open problem and is currently a point of intense research due to some recent advances in similar problems. This question is one of the Millenium Problems, for which there is a one-million dolar prize.
My interest, however, is mostly devoted to building a mathematical foundation for rigorous results related to turbulence. This comprises
looking for estimates of turbulence-related time-averaged mean quantities for Leray-Hopf weak solutions of the iNSE; developing and analysing a proper framework for statistical solutions of the iNSE as a rigorous way of assessing ensemble averages of the flow; and numerically investigating the results associated with the previous two items.
Representation of a push-forward of an initial measure by a semigroup (top); a phase-space statistical solution (middle); and a trajectory statistical solution (bottom).
This work stemmed from the study of statistical solutions of the Navier-Stokes. It aims towards a generalization of this notion to various types of evolution equations.
An abstract framework is formulated in which the phase-space of the evolutionary system is assumed to be a Hausdorff topological space, and the evolution equation is considered in a weak sense, in the dual of a topological vector space.
The main results obtained so far in this context are
the existence of solutions to the associated initial value problem under simple and natural conditions;
a vast number of applications showing the applicability of the theory; and
conditions for the convergence of subnets of statistical solutions depending on parameters of a family of equations.
There are challenging functional-analytic and measure-theoretic problems involved in this line of research due to the high degree of abstraction and the need to be as applicable as possible.
The "Dynamic Sugar Loaf", made of a heteroclinic cycle, a repeller focus as the Sun, and water waves completing the picture of a famous landmark in Rio de Janeiro.
Several real-life phenomena and advanced technological problems are modeled with systems involving evolutionary partial differential equations. Such systems appear in a multitude of areas, such as Physics, Engineering, Biology, Chemistry, Biochemistry, Finance, and so on.
As evolutionary partial differential equations, they may generate dynamical systems in phase spaces which are of infinite dimension. Many of these systems have very rich dynamics.
The study of evolutionary partial differential equations is thus of great interest and very challenging.
I started my research precisely in this area, working on it since my MSc at IM-UFRJ, continuing on it during my PhD at the Indiana University and thereafter. The study of evolutionary nonlinear partial differential equations as infinite-dimensional dynamical system was just taking off prior to my MSc and have been flourishing since then.
I have worked on several different types of nonlinear partial differential equations, from reaction-diffusion equations, to fluid-flow, dispersive, and wave-type equations. Currently, I have been working on fluid-flow models with viscous heating, important in some situations of magma flows and certain oil flows in pipelines.
Important questions in this field include
assessing the local well posedness of the system;
assessing the global well-posedness of the system and the existence of an associated dynamical system;
understanding the complexity of the associated dynamics;
analysing the possible finite-dimensionality of the asymptotic behavior of the system;
looking for finite-dimensional systems mimicking or approximating the dynamics of the system;
understanding the effect of the approximation of the infinite-dimensional system by finite-dimensional numerical schemes;
exploiting the finite dimensional asymptotic behavior of some system for control purposes; and so on.
Deep water prospection aspects
The salt layer, below, in yellow, modeled as a visco-elastic fluid, and being deformed into "mushroom"-like diapirs (geological intrusion), under the load of different layers of elastic sedimentary rocks.
Pre-salt oil became a fundamental asset for Brazil, but its exploitation is an extraordinary task.
The oil prospected before the pre-salt era lies in sedimentary layers at the bottom of the ocean or below ground. The pre-salt oil, on the other hand, lies within a carbonaceous layer, just below a thick 2km salt layer, which is below the sedimentary layers, and at the bottom of the ocean. The difficulties for prospection are enormous.
The aim of this project was initially to find a better model for the kinematics of the salt layer and then to study the behavior of the salt layer according to this model.
Salt rocks in the Earth's crust behave as solids on small time scales, but display a viscoelastic behavior on intermediate and long time scales. Some works assume a Newtonian viscous model or an Oldroyd-B visco-elastic model. We consider, on the other hand, a more general visco-elastic model proposed by I-Shi Liu (IM-UFRJ) that we termed the Mooney-Rivlin-Liu model.
Using experimental data obtained from a number of tests on samples of different types of salt rocks, we have showed that our model fits considerably better the observed deformations.
Having validated the model, we studied its analytical properties as well as some qualitive properties relevant for
the evolution of the salt layer in the Earth's crust;
the formation of salt diapirs flowing into the upper sediment layer;
the mechanical properties involved in the drilling process of boreholes through the salt layer; and
the optimization of the drilling path throught the different layers.
This is a joint project with CENPES/PETROBRÁS. We are now renewing this project to continue researching the properties of the salt layer and to include developing a model for the pre-salt layer and analysing the interaction between all those layers according to these models.
These diverse studies involve analysis of partial differential equations, dynamical systems, optimization methods, numerical analysis, and computer simulations.
Hop flowers and a plot of an experimental data of the humulone to iso-humulone conversion and a fitting by a saturation model.
There is a multitude of physical, chemical, and biological processes ocurring during the brewing and storage stages of beer manufacture.
Many of these processes can be given a mathematical formulation, and most of them can be modeled with systems of ordinary or partial differential equations. These are the sort of models that interest me the most (besides drinking the end product, of course).
From the physicochemical processes that turn the starch within the barley malt (one of the four main ingredients of beer) into the sugary wort that feeds the yeast (the most fundamental ingredient of beer); to the physicochemical process that turn the bitter and slightly soluble alpha-acids present in the hops (the third main ingredient of beer in this list) into the much more bitter and more soluble iso-alpha-acids that help balance the lingering sweetness that remains from the wort; to the biochemical processes involved when the yeast turns most of the sugar in the wort into alchohol and other aromatic compounds that make for a delicious (or not) beer. These are typical processes where ordinary differential equations can be used as models. (In case you notice anything missing, the fourth ingredient not mentioned yet is the most abundant one, namely water, which serves as the solvent for the other compounds.)
In most situations these processes can be assumed to occur within a spatially homogeneous and static medium, leading to systems of ordinary differential equations. Sometimes, however, the proccess requires the media to be treated as a flowing fluid, either as Newtonian-type flow inside a mash tun, a kettle, or a fermentation tank, or as a fluid flowing through a porous bed of spent grains, as in the sparging step. These processes lead to partial differential equations and are particularly important in the very design of the equipment, for efficiency (optimization) purposes.
Level sets of vorticity in a periodic, two-mode-forced incompressible 2D Navier-Stokes pseudo-spectrally simulated flow. (Click here for an extended simulation))
Turbulence is a fascinating phenomenom. It occurs in many types of fluids under various conditions. From the most visible ones, as in the formation of currents in rivers, to more hidden ones, as the flow of blood and other fluids in our bodies, of oil and gas in pipelines and wellbores, and of magma in the Earth's mantle and in magma chambers.
Despite the fact that there are well-accepted mathematical models to describe the motion of the associated fluid flows, explicit solutions to the modeling systems of equations exist only in very particular cases, so that alternative routes are needed.
Numerical solutions are largely exploited in engineering problems. However, in the case of turbulent flows, the complexity of the flow is such that the required mesh resolution for numerically modelling the dynamics of all the energetic spatial and temporal structures of the flow is impracticle for the current and even forthcoming computational power. Hence, appropriate simplifications and lower dimensional models must still be used. For this reason, research in turbulence is still a very active field.
My research mainly focuses on the system of equations known as the homogeneous and incompressible Navier-Stokes equations (iNSE), which is a model for the flow of certain Newtonian fluids such as water and oil, under most conditions. In a sense, this is the simplest possible type of fluid, but which already presents many theoretical and practical challenges.
In particular, the question of well-posedness of the iNSE is still an open problem and is currently a point of intense research due to some recent advances in similar problems. This question is one of the Millenium Problems, for which there is a one-million dolar prize.
My interest, however, is mostly devoted to building a mathematical foundation for rigorous results related to turbulence. This comprises
looking for estimates of turbulence-related time-averaged mean quantities for Leray-Hopf weak solutions of the iNSE; developing and analysing a proper framework for statistical solutions of the iNSE as a rigorous way of assessing ensemble averages of the flow; and numerically investigating the results associated with the previous two items.
Representation of a push-forward of an initial measure by a semigroup (top); a phase-space statistical solution (middle); and a trajectory statistical solution (bottom).
This work stemmed from the study of statistical solutions of the Navier-Stokes. It aims towards a generalization of this notion to various types of evolution equations.
An abstract framework is formulated in which the phase-space of the evolutionary system is assumed to be a Hausdorff topological space, and the evolution equation is considered in a weak sense, in the dual of a topological vector space.
The main results obtained so far in this context are
the existence of solutions to the associated initial value problem under simple and natural conditions;
a vast number of applications showing the applicability of the theory; and
conditions for the convergence of subnets of statistical solutions depending on parameters of a family of equations.
There are challenging functional-analytic and measure-theoretic problems involved in this line of research due to the high degree of abstraction and the need to be as applicable as possible.
The "Dynamic Sugar Loaf", made of a heteroclinic cycle, a repeller focus as the Sun, and water waves completing the picture of a famous landmark in Rio de Janeiro.
Several real-life phenomena and advanced technological problems are modeled with systems involving evolutionary partial differential equations. Such systems appear in a multitude of areas, such as Physics, Engineering, Biology, Chemistry, Biochemistry, Finance, and so on.
As evolutionary partial differential equations, they may generate dynamical systems in phase spaces which are of infinite dimension. Many of these systems have very rich dynamics.
The study of evolutionary partial differential equations is thus of great interest and very challenging.
I started my research precisely in this area, working on it since my MSc at IM-UFRJ, continuing on it during my PhD at the Indiana University and thereafter. The study of evolutionary nonlinear partial differential equations as infinite-dimensional dynamical system was just taking off prior to my MSc and have been flourishing since then.
I have worked on several different types of nonlinear partial differential equations, from reaction-diffusion equations, to fluid-flow, dispersive, and wave-type equations. Currently, I have been working on fluid-flow models with viscous heating, important in some situations of magma flows and certain oil flows in pipelines.
Important questions in this field include
assessing the local well posedness of the system;
assessing the global well-posedness of the system and the existence of an associated dynamical system;
understanding the complexity of the associated dynamics;
analysing the possible finite-dimensionality of the asymptotic behavior of the system;
looking for finite-dimensional systems mimicking or approximating the dynamics of the system;
understanding the effect of the approximation of the infinite-dimensional system by finite-dimensional numerical schemes;
exploiting the finite dimensional asymptotic behavior of some system for control purposes; and so on.
Deep water prospection aspects
The salt layer, below, in yellow, modeled as a visco-elastic fluid, and being deformed into "mushroom"-like diapirs (geological intrusion), under the load of different layers of elastic sedimentary rocks.
Pre-salt oil became a fundamental asset for Brazil, but its exploitation is an extraordinary task.
The oil prospected before the pre-salt era lies in sedimentary layers at the bottom of the ocean or below ground. The pre-salt oil, on the other hand, lies within a carbonaceous layer, just below a thick 2km salt layer, which is below the sedimentary layers, and at the bottom of the ocean. The difficulties for prospection are enormous.
The aim of this project was initially to find a better model for the kinematics of the salt layer and then to study the behavior of the salt layer according to this model.
Salt rocks in the Earth's crust behave as solids on small time scales, but display a viscoelastic behavior on intermediate and long time scales. Some works assume a Newtonian viscous model or an Oldroyd-B visco-elastic model. We consider, on the other hand, a more general visco-elastic model proposed by I-Shi Liu (IM-UFRJ) that we termed the Mooney-Rivlin-Liu model.
Using experimental data obtained from a number of tests on samples of different types of salt rocks, we have showed that our model fits considerably better the observed deformations.
Having validated the model, we studied its analytical properties as well as some qualitive properties relevant for
the evolution of the salt layer in the Earth's crust;
the formation of salt diapirs flowing into the upper sediment layer;
the mechanical properties involved in the drilling process of boreholes through the salt layer; and
the optimization of the drilling path throught the different layers.
This is a joint project with CENPES/PETROBRÁS. We are now renewing this project to continue researching the properties of the salt layer and to include developing a model for the pre-salt layer and analysing the interaction between all those layers according to these models.
These diverse studies involve analysis of partial differential equations, dynamical systems, optimization methods, numerical analysis, and computer simulations.
Hop flowers and a plot of an experimental data of the humulone to iso-humulone conversion and a fitting by a saturation model.
There is a multitude of physical, chemical, and biological processes ocurring during the brewing and storage stages of beer manufacture.
Many of these processes can be given a mathematical formulation, and most of them can be modeled with systems of ordinary or partial differential equations. These are the sort of models that interest me the most (besides drinking the end product, of course).
From the physicochemical processes that turn the starch within the barley malt (one of the four main ingredients of beer) into the sugary wort that feeds the yeast (the most fundamental ingredient of beer); to the physicochemical process that turn the bitter and slightly soluble alpha-acids present in the hops (the third main ingredient of beer in this list) into the much more bitter and more soluble iso-alpha-acids that help balance the lingering sweetness that remains from the wort; to the biochemical processes involved when the yeast turns most of the sugar in the wort into alchohol and other aromatic compounds that make for a delicious (or not) beer. These are typical processes where ordinary differential equations can be used as models. (In case you notice anything missing, the fourth ingredient not mentioned yet is the most abundant one, namely water, which serves as the solvent for the other compounds.)
In most situations these processes can be assumed to occur within a spatially homogeneous and static medium, leading to systems of ordinary differential equations. Sometimes, however, the proccess requires the media to be treated as a flowing fluid, either as Newtonian-type flow inside a mash tun, a kettle, or a fermentation tank, or as a fluid flowing through a porous bed of spent grains, as in the sparging step. These processes lead to partial differential equations and are particularly important in the very design of the equipment, for efficiency (optimization) purposes.
Level sets of vorticity in a periodic, two-mode-forced incompressible 2D Navier-Stokes pseudo-spectrally simulated flow. (Click here for an extended simulation))
Turbulence is a fascinating phenomenom. It occurs in many types of fluids under various conditions. From the most visible ones, as in the formation of currents in rivers, to more hidden ones, as the flow of blood and other fluids in our bodies, of oil and gas in pipelines and wellbores, and of magma in the Earth's mantle and in magma chambers.
Despite the fact that there are well-accepted mathematical models to describe the motion of the associated fluid flows, explicit solutions to the modeling systems of equations exist only in very particular cases, so that alternative routes are needed.
Numerical solutions are largely exploited in engineering problems. However, in the case of turbulent flows, the complexity of the flow is such that the required mesh resolution for numerically modelling the dynamics of all the energetic spatial and temporal structures of the flow is impracticle for the current and even forthcoming computational power. Hence, appropriate simplifications and lower dimensional models must still be used. For this reason, research in turbulence is still a very active field.
My research mainly focuses on the system of equations known as the homogeneous and incompressible Navier-Stokes equations (iNSE), which is a model for the flow of certain Newtonian fluids such as water and oil, under most conditions. In a sense, this is the simplest possible type of fluid, but which already presents many theoretical and practical challenges.
In particular, the question of well-posedness of the iNSE is still an open problem and is currently a point of intense research due to some recent advances in similar problems. This question is one of the Millenium Problems, for which there is a one-million dolar prize.
My interest, however, is mostly devoted to building a mathematical foundation for rigorous results related to turbulence. This comprises
looking for estimates of turbulence-related time-averaged mean quantities for Leray-Hopf weak solutions of the iNSE; developing and analysing a proper framework for statistical solutions of the iNSE as a rigorous way of assessing ensemble averages of the flow; and numerically investigating the results associated with the previous two items.
Representation of a push-forward of an initial measure by a semigroup (top); a phase-space statistical solution (middle); and a trajectory statistical solution (bottom).
This work stemmed from the study of statistical solutions of the Navier-Stokes. It aims towards a generalization of this notion to various types of evolution equations.
An abstract framework is formulated in which the phase-space of the evolutionary system is assumed to be a Hausdorff topological space, and the evolution equation is considered in a weak sense, in the dual of a topological vector space.
The main results obtained so far in this context are
the existence of solutions to the associated initial value problem under simple and natural conditions;
a vast number of applications showing the applicability of the theory; and
conditions for the convergence of subnets of statistical solutions depending on parameters of a family of equations.
There are challenging functional-analytic and measure-theoretic problems involved in this line of research due to the high degree of abstraction and the need to be as applicable as possible.
The "Dynamic Sugar Loaf", made of a heteroclinic cycle, a repeller focus as the Sun, and water waves completing the picture of a famous landmark in Rio de Janeiro.
Several real-life phenomena and advanced technological problems are modeled with systems involving evolutionary partial differential equations. Such systems appear in a multitude of areas, such as Physics, Engineering, Biology, Chemistry, Biochemistry, Finance, and so on.
As evolutionary partial differential equations, they may generate dynamical systems in phase spaces which are of infinite dimension. Many of these systems have very rich dynamics.
The study of evolutionary partial differential equations is thus of great interest and very challenging.
I started my research precisely in this area, working on it since my MSc at IM-UFRJ, continuing on it during my PhD at the Indiana University and thereafter. The study of evolutionary nonlinear partial differential equations as infinite-dimensional dynamical system was just taking off prior to my MSc and have been flourishing since then.
I have worked on several different types of nonlinear partial differential equations, from reaction-diffusion equations, to fluid-flow, dispersive, and wave-type equations. Currently, I have been working on fluid-flow models with viscous heating, important in some situations of magma flows and certain oil flows in pipelines.
Important questions in this field include
assessing the local well posedness of the system;
assessing the global well-posedness of the system and the existence of an associated dynamical system;
understanding the complexity of the associated dynamics;
analysing the possible finite-dimensionality of the asymptotic behavior of the system;
looking for finite-dimensional systems mimicking or approximating the dynamics of the system;
understanding the effect of the approximation of the infinite-dimensional system by finite-dimensional numerical schemes;
exploiting the finite dimensional asymptotic behavior of some system for control purposes; and so on.
Deep water prospection aspects
The salt layer, below, in yellow, modeled as a visco-elastic fluid, and being deformed into "mushroom"-like diapirs (geological intrusion), under the load of different layers of elastic sedimentary rocks.
Pre-salt oil became a fundamental asset for Brazil, but its exploitation is an extraordinary task.
The oil prospected before the pre-salt era lies in sedimentary layers at the bottom of the ocean or below ground. The pre-salt oil, on the other hand, lies within a carbonaceous layer, just below a thick 2km salt layer, which is below the sedimentary layers, and at the bottom of the ocean. The difficulties for prospection are enormous.
The aim of this project was initially to find a better model for the kinematics of the salt layer and then to study the behavior of the salt layer according to this model.
Salt rocks in the Earth's crust behave as solids on small time scales, but display a viscoelastic behavior on intermediate and long time scales. Some works assume a Newtonian viscous model or an Oldroyd-B visco-elastic model. We consider, on the other hand, a more general visco-elastic model proposed by I-Shi Liu (IM-UFRJ) that we termed the Mooney-Rivlin-Liu model.
Using experimental data obtained from a number of tests on samples of different types of salt rocks, we have showed that our model fits considerably better the observed deformations.
Having validated the model, we studied its analytical properties as well as some qualitive properties relevant for
the evolution of the salt layer in the Earth's crust;
the formation of salt diapirs flowing into the upper sediment layer;
the mechanical properties involved in the drilling process of boreholes through the salt layer; and
the optimization of the drilling path throught the different layers.
This is a joint project with CENPES/PETROBRÁS. We are now renewing this project to continue researching the properties of the salt layer and to include developing a model for the pre-salt layer and analysing the interaction between all those layers according to these models.
These diverse studies involve analysis of partial differential equations, dynamical systems, optimization methods, numerical analysis, and computer simulations.
Hop flowers and a plot of an experimental data of the humulone to iso-humulone conversion and a fitting by a saturation model.
There is a multitude of physical, chemical, and biological processes ocurring during the brewing and storage stages of beer manufacture.
Many of these processes can be given a mathematical formulation, and most of them can be modeled with systems of ordinary or partial differential equations. These are the sort of models that interest me the most (besides drinking the end product, of course).
From the physicochemical processes that turn the starch within the barley malt (one of the four main ingredients of beer) into the sugary wort that feeds the yeast (the most fundamental ingredient of beer); to the physicochemical process that turn the bitter and slightly soluble alpha-acids present in the hops (the third main ingredient of beer in this list) into the much more bitter and more soluble iso-alpha-acids that help balance the lingering sweetness that remains from the wort; to the biochemical processes involved when the yeast turns most of the sugar in the wort into alchohol and other aromatic compounds that make for a delicious (or not) beer. These are typical processes where ordinary differential equations can be used as models. (In case you notice anything missing, the fourth ingredient not mentioned yet is the most abundant one, namely water, which serves as the solvent for the other compounds.)
In most situations these processes can be assumed to occur within a spatially homogeneous and static medium, leading to systems of ordinary differential equations. Sometimes, however, the proccess requires the media to be treated as a flowing fluid, either as Newtonian-type flow inside a mash tun, a kettle, or a fermentation tank, or as a fluid flowing through a porous bed of spent grains, as in the sparging step. These processes lead to partial differential equations and are particularly important in the very design of the equipment, for efficiency (optimization) purposes.
Calculus of one and several variables
Linear algebra
Differential equations
Mathematical modeling
Stochastic differential equations
Advanced linear algebra
Real Analysis
Complex variables
Linear Algebra
Measure theory
Functional analysis
Semigroups
Navier-Stokes equations and turbulence
Stochastic differential equations
"Resultados recentes sobre as equações de Navier-Stokes para fluidos incompressíveis". Primeira Escola Brasileira de Equações Diferenciais (I EBED), IMECC-Unicamp, Campinas, SP. June 9-13, 2003. Slides
"Equações de Navier-Stokes e turbulência". Summer Program at LNCC, Petrópolis, RJ. February 24-27, 2003. Slides
"Equações de Navier-Stokes e turbulência". Summer Program at IM-UFRJ, Rio de Janeiro, RJ. January 4, 6, 8, 18, 20, 22, 2002.
"Navier-Stokes equations and the statistical theory of turbulence", Seventh Paseky School on Mathematical Theory in Fluid Mechanics, Paseky, Czech Republic. June 3-10, 2001.
"Sistemas dinâmicos em dimensão infinita". Short course at IM-UFRJ, Rio de Janeiro, RJ. July 1995.
Several undergraduate students
Seven MSC students
Two PhD students:
Fábio Antônio Tavares Ramos, on "Statistical estimates and dissipation anomaly in fluid turbulence", August 3, 2007
Cecília Freire Mondaini, on "Statistical solutions of the Navier-Stokes equations", May 20, 2014
Anne Caroline Bronzi, "Statistical solutions of the Navier-Stokes equations", August 2010 to July 2012
Calculus of one and several variables
Linear algebra
Differential equations
Mathematical modeling
Stochastic differential equations
Advanced linear algebra
Real Analysis
Complex variables
Linear Algebra
Measure theory
Functional analysis
Semigroups
Navier-Stokes equations and turbulence
Stochastic differential equations
"Resultados recentes sobre as equações de Navier-Stokes para fluidos incompressíveis". Primeira Escola Brasileira de Equações Diferenciais (I EBED), IMECC-Unicamp, Campinas, SP. June 9-13, 2003. Slides
"Equações de Navier-Stokes e turbulência". Summer Program at LNCC, Petrópolis, RJ. February 24-27, 2003. Slides
"Equações de Navier-Stokes e turbulência". Summer Program at IM-UFRJ, Rio de Janeiro, RJ. January 4, 6, 8, 18, 20, 22, 2002.
"Navier-Stokes equations and the statistical theory of turbulence", Seventh Paseky School on Mathematical Theory in Fluid Mechanics, Paseky, Czech Republic. June 3-10, 2001.
"Sistemas dinâmicos em dimensão infinita". Short course at IM-UFRJ, Rio de Janeiro, RJ. July 1995.
Several undergraduate students
Seven MSC students
Two PhD students:
Fábio Antônio Tavares Ramos, on "Statistical estimates and dissipation anomaly in fluid turbulence", August 3, 2007
Cecília Freire Mondaini, on "Statistical solutions of the Navier-Stokes equations", May 20, 2014
Anne Caroline Bronzi, "Statistical solutions of the Navier-Stokes equations", August 2010 to July 2012
