Analytic Gelfand-Shilov smoothing effect of the spatially homogeneous Landau equation

TL;DR

This paper proves that solutions to the spatially homogeneous Landau equation with hard potential become analytic in Gelfand-Shilov spaces over time, demonstrating a smoothing effect similar to heat diffusion.

Contribution

It establishes the Gelfand-Shilov regularizing effect for solutions of the Landau equation in a close-to-equilibrium setting, a novel analytic smoothing result.

Findings

Solutions become analytic in Gelfand-Shilov space for positive time

Analytic radius evolution mimics heat equation behavior

Regularity enhancement from initial $L^2$ data

Abstract

In this work, we study the nonlinear spatially homogeneous Landau equation with hard potential in a close-to-equilibrium framework, we show that the solution to the Cauchy problem with $L^2$ initial datum enjoys a analytic Gelfand-Shilov regularizing effect in the class $S^1_1(\mathbb{R}^3)$, meaning that the solution of the Cauchy problem and its Fourier transformation are analytic for any positive time, the evolution of analytic radius is similar to the heat equation.