Long time behavior of temperature gradient driven compressible fluid flows

• Eduard Feireisl

Room: Salle de conférence 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 sets, asymptotic compactness of global trajectories, stationary statistical solutions, and, last but not the least, the problem of convergence to equilibria.

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

Room: Salle de conférence 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.

Phenomenology of fluid turbulence and its stochastic representation

• Laurent Chevillard

Room: Salle de conférence 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 precise limit that I will try to define), which could be viewed as a concise model of the turbulent velocity field. Joint work with C. Garban, R. Pereira, R. Rhodes, R. Robert and V. Vargas.

Stability of Rayleigh-Jeans equilibria in the kinetic FPUT equation

• Angeliki Menegaki

Room: Salle de conférence 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 equilibrium solutions. This is a joint work with Pierre Germain (Imperial College London) and Joonhyun La (KIAS).

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

• Philippe Bonneton

Room: Salle de conférence 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 similarities between sawtooth waves in the inner surf zone and Burgers wave solutions. This theoretical spectrum shows very good agreement with laboratory experiments covering a wide range of incident random wave conditions.

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

• Anne-Sophie de Suzzoni

Room: Salle de conférence 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 same invariance as the Schrödinger equation, and I will show that in this example, exact resonances (always) take precedence over quasi-resonances, so that the effective dynamics of the statistical evolution of the solutions are not kinetic. However, these dynamics are not linear (let alone trivial). I will present the problem and the ideas involved in deriving the effective dynamics and some elements of proof: in particular, I will describe the representation of solutions of the initial equation in diagrammatic form. This talk is based on a joint work with Annalaura Stingo (X) and Arthur Touati (Bordeaux).

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

• Ludovic Godard-Cadillac

Room: Salle de conférence 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 critical case s=1/2, we introduced a weaker notion of solution (V-weak solutions) to give a meaning to the equation and prove global existence.

Scattering, random phase and wave turbulence

• Erwan Faou

Room: Salle de conférence 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 real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converges towards the kinetic operator but with different form of convergence.

Uniform resolvent estimates and smoothing effects related to Heisenberg sublaplacians

• Luz Roncal

Room: Salle de conférence 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 that it fails to be dispersive, as shown by Bahouri, Gérard, and Xu. In fact, Strichartz or L^p−L^q estimates cannot hold in general. In this talk we will discuss uniform resolvent estimates on the Heisenberg group and their application to obtain certain smoothing effects for Schrödinger equations. Joint work with Luca Fanelli, Haruya Mizutani, and Nico Michele Schiavone.

Linear turbulence

• Geoffrey Beck

Room: Salle de conférence 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 phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a wave equation, which corre- sponds to a wave operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time. The proposed linear dynamics can be generalized to more general spectra, possibly non-radial, including sea wavenumber spectra such as the JONSWAP spectrum. We apply a finite volume method in the Fourier variables formulation in order to reach the invariant measure of the equation.

Energy cascades and condensation via coherent dynamics in Hamiltonian systems

• Anxo Farina Biasi

Room: Salle de conférence 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 exhibit power-law spectrum formation in finite time, leading to the blow-up of Sobolev norms and singularities