ICTS

We start by demonstrating that numerical methods do not necessarily converge to entropy or admissible weak solutions of the Euler and Navier-Stokes equations of fluid dynamics on mesh refinement due to appearance of eddies at smaller and smaller scales. As an alternative, we revisit the concept of statistical solutions which are time-parametrized probability measures, consistent with the fluid evolution. We empirically show that the same numerical methods converge to a statistical solution and also derive verifiable sufficient conditions under which this convergence can be made rigorous. Numerical experiments illustrating interesting properties of statistical solutions are also presented. We conclude by showing how state of the art generative AI models (conditional diffusion) can significantly lower the cost of computing statistical solutions while maintaining accuracy.

We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.

In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.

Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.

There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.

We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.

In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.

Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.

There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.

In the second part, I will discuss recent developments on stochastic PDEs of fluid mechanics such as probabilistically strong solution, uniqueness path-wise or in law, non-uniqueness of Markov selection, as well as some related conjectures such as those of Onsager and Taylor.

There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.

In the third part of this talk, I will discuss recent developments on singular stochastic PDEs. In short, these are SPDEs for which the random noise acting as a force is so rough that the nonlinear term becomes ill-defined as a product of distributions. We will discuss solution theory, local and global, as well as recent non-uniqueness results.

We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.

In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transport model, introduced by Kraichnan, has been widely studied due to its peculiar features, like spontaneous stochasticity. In the case of incompressible 2D Euler equations, we show that the addition of nonsmoooth Kraichnan noise improves significantly the well-posedness theory, for example bringing uniqueness among finite enstrophy solutions. If time allows, we show further regularization properties of the Kraichnan noise.

Based on classical works (e.g. the works by Flandoli and coauthors) and on joint works with Marco Bagnara, Michele Coghi, Lucio Galeati, Francesco Grotto.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.

However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.

Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.

However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.

Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.

However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.

Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.

Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-Barat in his PhD thesis dedicated to the partially dissipative case, and in J.-P. Adogbo’s thesis dedicated to the partially diffusive case. Compared to the previous works, the authors adopted a `critical’ Besov framework that allows not only to consider a larger class of data but also to prove by very simple scaling arguments optimal time-decay estimates and to justify the strong relaxation or diffusive limit.
In this course, we will provide the basics of the Fourier analysis and nonlinear analysis allowing to introduce the functional framework that is needed to establish the aforementioned results, then we will investigate partially dissipative or diffusive systems. Some examples of applications in fluid dynamics will be given.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-Barat in his PhD thesis dedicated to the partially dissipative case, and in J.-P. Adogbo’s thesis dedicated to the partially diffusive case. Compared to the previous works, the authors adopted a `critical’ Besov framework that allows not only to consider a larger class of data but also to prove by very simple scaling arguments optimal time-decay estimates and to justify the strong relaxation or diffusive limit.
In this course, we will provide the basics of the Fourier analysis and nonlinear analysis allowing to introduce the functional framework that is needed to establish the aforementioned results, then we will investigate partially dissipative or diffusive systems. Some examples of applications in fluid dynamics will be given.

The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.

However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume preserving under natural assumptions on the noise and vector fields.

Returning to fluid equations, we shall use the above results to prove well-posedness of the roughly perturbed Euler equation in Lagrangian form when d=2. We will consider well-posedness in the class of equations with bounded and integrable vorticity, also known as Yudovich theory.