Small Knudsen rate of convergence to rarefaction wave for the Landau equation

TL;DR

This paper proves the convergence of solutions of the Landau equation with Coulomb potentials to rarefaction waves of the Euler system as the Knudsen number approaches zero, with explicit convergence rates and uniform-in-epsilon solutions.

Contribution

It constructs unique global solutions near a Maxwellian for small Knudsen number and establishes their convergence to rarefaction waves with explicit rates, under a specific scaling.

Findings

Established uniform-in-epsilon solutions around rarefaction waves.

Proved convergence rate of solutions to the Euler rarefaction wave as epsilon tends to zero.

Used refined energy methods under a specific scaling transformation.

Abstract

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number $\epsilon>0$ is vanishing. Precisely, whenever $\epsilon>0$ is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter $a\in [\frac{2}{3},1]$, we construct the unique global-in-time uniform-in-$\epsilon$ solution around a local Maxwellian whose fluid quantities are the rarefaction wave of the corresponding Euler system. In the meantime, we establish the convergence of solutions to the Riemann rarefaction wave uniformly away from $t=0$ at a rate $\epsilon^{\frac{3}{5}-\frac{2}{5}a}|\ln \epsilon|$ as $\epsilon\to 0$. The proof is based on the refined energy approach combining [19] and [32] under the scaling transformation $(t,x)\to (\epsilon^{-a}t,\epsilon^{-a}x)$.