Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

TL;DR

This paper proves that solutions to the nonlinear Landau equation of Maxwellian molecules become analytic in both space and velocity variables over time, demonstrating a smoothing effect for the first time in such kinetic equations.

Contribution

It establishes the first analytic smoothing effect for the spatially inhomogeneous nonlinear Landau equation using microlocal analysis and energy estimates.

Findings

Global solutions become analytic in space and velocity for any positive time.

Small initial perturbations in H^r_x(L^2_v) norm lead to unique global solutions.

The method involves a novel time integral weight linked to the kinetic transport operator.

Abstract

We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if $H^r_x(L^2_v), r >3/2$ norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position $x$ and velocity $v$ variables for any time $t>0$. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight associated with the kinetic transport operator.