Conference on Harmonic Analysis and Turbulence

Liste des Contributions

1. Long time behavior of temperature gradient driven compressible fluid flows

Eduard Feireisl

14/04/2025 14:00

We discuss several recent results concerning the qualitative properties of global in time weak solutions to the Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluids. In particular, we focus on the Rayleigh-Benard convection problem, where the fluid motion is driven by the temperauture gradient. We discuss the existence of bounded absorbing...

2. What are the optimal conditions on the domains and the operator to ensure the solvability of boundary values problems for elliptic operators with data in L^p?

Joseph Feneuil

14/04/2025 15:00

In this talk, I will discuss the properties of the Dirichlet boundary value problems with data in L^p and W^{1,p} and present the current results on their solvability. I will then address the challenges of extending these results to the Neumann problem and share our progress in this area.

3. Phenomenology of fluid turbulence and its stochastic representation

Laurent Chevillard

15/04/2025 09:00

I will be presenting/recalling some key ingredients of the phenomenology of three-dimensional fluid turbulence, which concerns the statistical behavior of the solutions of the forced Navier-Stokes equations, as they are observed in laboratory and natural flows. Then, I will propose a random vector field, statistically homogeneous and isotropic, incompressible, asymptotically rough (in a...

4. Stability of Rayleigh-Jeans equilibria in the kinetic FPUT equation

Angeliki Menegaki

15/04/2025 10:30

In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from the microscopic Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the global existence and stability of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic...

5. Analogy between wave dynamics in the nearshore inner surf zone and Burgers turbulence

Philippe Bonneton

15/04/2025 11:30

In this presentation, we investigate the spectral behavior of random sawtooth waves propagating in the nearshore inner surf zone. We show that the elevation energy spectrum exhibits a universal shape, following a $\omega^2$ trend in the inertial subrange and an exponential decay in the diffusive subrange (where $\omega$ is the angular frequency). A theoretical spectrum is derived based on the...

6. Trivial resonances for a system of Klein-Gordon equations and statistical applications

Anne-Sophie de Suzzoni

15/04/2025 14:00

In the derivation of the wave kinetic equation coming from the Schrödinger equation, a key feature is the invariance of the Schrödinger equation under the action of U(1). This allows quasi-resonances of the equation to drive the effective dynamics of the statistical evolution of solutions to the Schrödinger equation. In this talk, I will give an example of an equation that does not have the...

7. Existence and Uniqueness for the SQG Vortex-Wave System when the Vorticity is Constant near the Point-Vortex

Ludovic Godard-Cadillac

15/04/2025 15:00

The aim of this talk is to study the Cauchy theory for the vortex-wave system associated to the Surface Quasi-Geostrophic equation with parameter 0<s<1. We obtain local existence of classical solutions in H^4 under the standard "plateau hypothesis'', H^2-stability of the solutions, and a blow-up criterion. In the sub-critical case s>1/2 we establish global existence of weak solutions. For the...

11. Scattering, random phase and wave turbulence

Erwan Faou

15/04/2025 16:30

We start from the remark that in wave turbulence theory, exemplified by the cubic twodimensional Schrödinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the...

8. Uniform resolvent estimates and smoothing effects related to Heisenberg sublaplacians

Luz Roncal

16/04/2025 09:00

Uniform resolvent estimates play a fundamental role in the study of spectral and
scattering theory for Schr¨odinger equations. In particular, they are closely connected
to global-in-time dispersive estimates, such as Strichartz estimates. In contrast with the Euclidean setting, a peculiar fact of the Schrödinger evolution equation associated to the sublaplacian on the Heisenberg group is...

9. Linear turbulence

Geoffrey Beck

16/04/2025 10:30

Wave turbulence shares three key characteristics with hydrodynamic turbulence: multiple scales, randomness and the presence of cascades. Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. With C.-E. Bréhier, L. Chevillard, I. Gallagher, R. Grande and W. Ruffenach, we have constructed a linear equation that mimics the...

10. Energy cascades and condensation via coherent dynamics in Hamiltonian systems

Anxo Farina Biasi

16/04/2025 11:30

In this talk, I will present recent results on energy cascades and structure formation in Hamiltonian systems. I will introduce two families of solvable systems that explicitly illustrate the dynamical development of energy cascades and the emergence of large- and small-scale structures. Some solutions represent condensate formation through highly coherent dynamics, while all cascade solutions...