Cauchy problem for the spatially homogeneous landau equation with shubin class initial datum and gelfand-shilov smoothing effect

TL;DR

This paper analyzes the nonlinear spatially homogeneous Landau equation with initial data in Shubin space, demonstrating spectral properties and proving that solutions become ultra-analytic with exponential decay over time.

Contribution

It introduces a spectral analysis approach to the Landau equation with Shubin class initial data and proves Gelfand-Shilov smoothing effects for solutions.

Findings

Existence of weak solutions for initial data in Shubin space.

Spectral decomposition shows the near-linearity of the Landau operator.

Solutions exhibit ultra-analyticity and exponential decay over time.

Abstract

In this work, we study the nonlinear spatially homogeneous Lan-dau equation with Maxwellian molecules, by using the spectral analysis, we show that the non linear Landau operators is almost linear, and we prove the existence of weak solution for the Cauchy problem with the initial datum belonging to Shubin space of negative index which conatins the probability measures. Based on this spectral decomposition, we prove also that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy problem with Shubin class initial datum is ultra-analytics and exponential decay for any positive time.