  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.
