Analytic smoothing effect of the time variable for the spatially homogeneous Landau equation

TL;DR

This paper proves that solutions to the spatially homogeneous Landau equation with hard potentials become analytic in time, similar to heat equation smoothing, starting from L2 initial data.

Contribution

It demonstrates the analytic regularizing effect of the time variable for the Landau equation with hard potentials in a close-to-equilibrium setting.

Findings

Solutions become analytic in time for positive times.

The smoothing effect matches that of the heat equation.

Valid for initial data in L2 space.

Abstract

In this work, we study the Cauchy problem of the spatially homogeneous Landau equation with hard potentials in a close-to-quilibrium framework. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time. So that the smoothing effect of Cauchy problem for the spatially homogeneous Landau equation with hard potentials is exactly same as heat equation.