Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials

TL;DR

This paper demonstrates that solutions to the spatially inhomogeneous Landau equations with hard potentials exhibit an analytic smoothing effect in both space and velocity, akin to hypoelliptic Fokker-Planck operators, using a novel time-average operator.

Contribution

It introduces a new time-average operator and establishes the analytic smoothing effect for low-regularity solutions of the Landau equations, linking them to hypoelliptic Fokker-Planck behavior.

Findings

Solutions become analytic in space and velocity variables.

The Landau equations behave like hypoelliptic Fokker-Planck operators.

A new time-average operator is key to the proof.

Abstract

We study the spatially inhomogeneous Landau equations with hard potential in the perturbation setting, and establish the analytic smoothing effect in both spatial and velocity variables for a class of low-regularity weak solutions. This shows the Landau equations behave essentially as the hypoelliptic Fokker-Planck operators. The spatial analyticity relies on a new time-average operator, and the proof is based on a straightforward energy estimate with a careful estimate on the derivatives with respect to the new time-average operator.