A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation

TL;DR

This paper demonstrates that solutions to the spatially homogeneous Landau equation near equilibrium become ultra-analytic over time, showing a strong regularizing effect in both physical and Fourier space.

Contribution

It establishes the Gelfand-Shilov regularizing effect for the Landau equation near Maxwellian equilibrium, highlighting ultra-analyticity of solutions.

Findings

Solutions gain ultra-analytic regularity for any positive time

Both the fluctuation and its Fourier transform become ultra-analytic

The regularizing effect applies in a close-to-equilibrium setting

Abstract

We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect implying the ultra-analyticity of both the fluctuation and its Fourier transform for any positive time.