Program

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 talk about some recent work on the problem of establishing rigorously a wave turbulence theory for water waves systems. This is a classical problem in Mathematical Physics, going back to pioneering work of Hasselmann. More precisely, I will discuss recent results on the long-time existence of solutions of water waves systems and on the well-posedness theory of the associated kinetic equations. To address the quasi-linear nature of the problems we propose a new mechanism, based on a combination of two main ingredients: (1) deterministic energy estimates for all solutions that are small in $L^\infty$-based norms, and (2) probabilistic arguments aimed at understanding propagation of randomness on long time intervals. This is joint work with Yu Deng and Fabio Pusateri.

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.

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 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.