A remark about time-analyticity of the linear Landau equation with soft potential

TL;DR

This paper proves that solutions to the linear spatially homogeneous Landau equation with soft potentials become analytic in time for positive times, demonstrating a heat-like smoothing effect starting from L2 initial data.

Contribution

It establishes the time-analytic regularizing effect for the Landau equation with soft potentials, showing smoothing similar to the heat equation.

Findings

Solutions become analytic in time for positive times.

The smoothing effect is similar to that of the heat equation.

The result holds for L2 initial data.

Abstract

In this note, we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials. 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 linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.