Ultra-analytic effect of Cauchy problem for a class of kinetic equations

TL;DR

This paper investigates the ultra-analytic smoothing effects of the Cauchy problem for certain kinetic equations, demonstrating optimal regularity results similar to heat equations for both homogeneous and inhomogeneous cases.

Contribution

It provides new optimal regularity results for the Cauchy problem of kinetic equations, including the Landau and Fokker-Planck equations, highlighting ultra-analytic effects.

Findings

Optimal ultra-analytic smoothing effects for homogeneous Landau and Fokker-Planck equations.

Analytic smoothing effects for inhomogeneous linear Landau equation.

Results are comparable to heat equation regularity.

Abstract

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous non linear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.