Nonlinear chaotic Vlasov equations

TL;DR

This paper investigates nonlinear Vlasov equations on chaotic manifolds, demonstrating global solutions that exponentially converge to equilibrium using advanced microlocal analysis techniques.

Contribution

It introduces refined microlocal anisotropic Sobolev spaces tailored for nonlinear Vlasov equations on Anosov manifolds, advancing the understanding of their long-term behavior.

Findings

Existence of global solutions for small initial data

Exponential convergence to equilibrium

Potential strongly converges to zero

Abstract

In this article, we study nonlinear Vlasov equations with a smooth interaction kernel on a compact manifold without boundary where the geodesic flow exhibits strong chaotic behavior, known as the Anosov property. We show that, for small initial data with finite regularity and supported away from the null section, there exist global solutions to the nonlinear Vlasov equations which weakly converge to an equilibrium of the free transport equation, and whose potential strongly converges to zero, both with exponential speed. Central to our approach are microlocal anisotropic Sobolev spaces, originally developed for studying Pollicott-Ruelle resonances, that we further refine to deal with the geometry of the full cotangent bundle, which paves the way to the analysis of nonlinear Vlasov equations.