# Smoothing effects and decay estimate of the solution of the linearized two species Landau equation

## TL;DR

This paper investigates the linearized two-species Landau equation, demonstrating instant smoothing effects and analyzing the long-term decay of solutions without requiring initial smoothness.

## Contribution

It establishes the instant regularization of solutions and provides decay estimates for the linearized two-species Landau equation with minimal initial smoothness assumptions.

## Key findings

- Solutions become smooth instantly in space and momentum variables.
- Decay estimates for solutions are obtained over large times.
- No initial smoothness assumptions are needed for the results.

## Abstract

We study the Landau equation for a mixture of two species in the whole space, with initial condition of one species near a vacuum and the other near a Maxwellian equilibrium state. For the linearized level, without any smoothness assumption on the initial data, it is shown that the solution becomes smooth instantaneously in both the space and momentum variables. Moreover, the large time behavior of the solution is also obtained.