Program

We study the top Lyapunov exponent of stochastic Navier stokes and prove quantitative lower bounds on it. The study is linked to the ergodic properties of a Markov process on projective space, for which we obtain unique ergodicity. The proof rests on low mode recurrence, Malliavin controllability in projective space, and an adaptation of asymptotic strong Feller techniques. This is based on joint work with M. Hairer, S. Punshon-Smith and J. Yi, and on work in progress with S. Punshon-Smith.

Turbulent fluid flows are described by the Navier-Stokes equations, whose inviscid limit formally leads to the Euler equations. While viscosity tends to zero, solutions may develop increasingly fine structures, raising fundamental questions about predictability and uniqueness. The evolution of infinitesimal perturbations in statistically stationary turbulent flows is analyzed by direct numerical simulations. Small errors do not remain confined to small scales: instead, they propagate toward large scales through an inertial mechanism analogous to Richardson dispersion. This growth appears to be independent of the details of energy injection, suggesting a universal phenomenon governed by the turbulent cascade itself. Such intrinsic and possibly universal spontaneous stochasticity in statistically stationary turbulence may provide a dynamical perspective of the potential non-uniqueness of weak Euler solutions in the inviscid limit, and raises the question of whether a statistical selection principle for physically relevant solutions may emerge.

In this talk, we summarize recent progress made in the understanding of the well-posedness and regularity of hard-potential kinetic equations through the means of collisional averaging. We will present results related to the global well-posedness of the space inhomogeneous 4-wave KWE,  a well-posedness/illposedness threshold for quasilinear KWE of MMT-type, as well as moment preserving weighted Young’s inequalities for the Boltzmann gain operator.

In this talk, I will discuss several fundamental problems in wave turbulence that arise in physics and astrophysics, including inertial waves, odd waves, and plasma waves. A unifying feature of these systems is anisotropy, with energy cascades shaped by external forcing or constraints. Despite the additional complexity introduced by anisotropy, it remains possible to derive the wave kinetic equation and obtain its solutions.

In a 2009 paper "Turbulence in Integrable Systems", Zakharov discussed weakly integrable systems which have an infinite, but not sufficient for the full integrability, number of invariants. An example is KP1 for which Zakharov wrote a kinetic equation describing wave turbulence, and which has an infinite number of invariants and thermodynamic equilibria. In the present talk, I will report on some rigorous results and  numerical simulations of this kinetic equation, in particular, a kinetic instability of a wavepacket, and relaxation to a thermal equilibrium characterised by a condensation into low-frequency waves.

I will discuss some recent computer-assisted work on the question of non-uniqueness of solutions of the incompressible Navier-Stokes in 3D. The incompressible Navier-Stokes equations are "regular" equations in the class of axi-symmetric swirl-free solutions, for which large data global regularity of smooth solutions is known. The point of the lecture is to provide evidence that non-uniqueness of solutions still holds in critical and slightly super-critical spaces.

We discuss the wave turbulence properties of the Fermi-Pasta-Ulam-Tsingou system for coupled harmonic oscillators. While it was initially predicted that the system should exhibit thermalization, numerical experiments showed quasi-periodic behavior, giving rise to the so-called ``FPUT paradox". In this talk, we discuss the behavior of the system in a weakly nonlinear regime in the kinetic limit (where there is equipartition of energy). In particular, we derive the wave kinetic equation for the $\beta$-FPUT. Due to the 1D nature of the system, the derivation is only currently possible at subcritical times due to combinatorial issues present in 1D. Additional difficulties present in this system are the non-resonant terms in the nonlinearity as well as the need for a time-dependent phase renormalization to remove divergent interactions.

In this talk, we discuss the kinetic theory of one-dimensional nonlinear oscillator chains, of which the most famous example is the Fermi-Pasta-Ulam-Tsingou equation.

Intermittency is a remarkable and robust feature of three-dimensional turbulence for which we still lack explanation from first principles. It will be shown how a dissipation with a non-trivial lower-dimensional part induces a quantitative intermittent regularity on the weak solution.

I will present some results on large deviations in stochastic binary collision models. I will show that, although the dynamics conserves energy, atypical paths that violate energy conservation can occur with exponentially small probability as the number of particles increases. As a consequence, a dynamical phase transition may arise in the asymptotic behavior of the total number of collisions. Specifically, I will discuss examples related to energy transport and the role of anomalous diffusion in these processes.

I will present recent work on the derivation of an effective dynamics for the correlations associated to a semilinear Klein-Gordon system with random initial data. Due to the lack of invariances of this system, the effective dynamics is led by the trivial resonances and is not of kinetic type. I will motivate and present the model and the main result, and give some elements of the proof. This is a joint work with Anne-Sophie de Suzzoni (Evry) and Annalaura Stingo (Ecole polytechnique).

The theory of wave turbulence, which started in the 1920s as the wave analog of Boltzmann’s kinetic theory, has been an active field of physics in the last century, with substantial applications in science. In this talk I will review some recent works, joint with Zaher Hani, that establish the rigorous mathematical foundation of this subject. In particular, we present the justification of the wave kinetic equation up to arbitrarily large kinetic time, which is the first long time result ever obtained in any nonlinear kinetic limit.

A review on the history of mathematical physics of Boltzmann equation, classical and recent results.

In this talk, I will consider a particular class of non-equilibrium solutions to the Boltzmann equation, known as homoenergetic solutions. These solutions are useful for modeling the dynamics of Boltzmann gases subjected to mechanical deformations (such as shear, expansion, or compression), thereby yielding insight into the behavior of open systems. Their analytical significance lies in the fact that they model systems driven genuinely far from equilibrium: because they lack a detailed balance condition, their long-time behavior is not governed by a classical Maxwellian distribution, and the landscape of possible long-time asymptotics is extremely rich and diverse. I will present different possible long-time asymptotics for homoenergetic solutions to the Boltzmann equation in the case of shear deformations, and discuss some conjectures and open problems in this direction.

Fisher and Kolmogorov-Petrovsky-Piskunov (1937) introduced a fundamental reaction-diffusion model for the interaction between linear diffusive behavior and a nonlinear birth-death growth process in the context of population dynamics. They proved the existence of traveling waves above a certain minimal speed. The equation can be reformulated in terms of branching Brownian motion, and the probabilist Bramson later (1978) localized these traveling waves more precisely by establishing a logarithmic correction to the linear speed of propagation. It is possible to provide a finer-scale description underlying the diffusion/noise with a kinetic model of the transport-reaction type: the kinetic Fisher-KPP equation. Bouin, Calvez, and Nadin (2014) established the existence of traveling waves (when velocities are bounded), and we prove the Bramson correction for these traveling waves. The proof relies on half-space hypocoercivity estimates and the PDE method of Hamel, Nolen, Roquejoffre, and Ryzhik. This is a two-part joint work with Bouin and Mischler.