Landau方程向稀疏波的小Knudsen收敛速率
Small Knudsen rate of convergence to rarefaction wave for the Landau equation
摘要：
本文研究了当Knudsen数$ε>0$为零时，带库仑势的Landau方程可压缩Euler方程组对稀疏波的流体动力学极限.精确地说，当$ε>0$很小时，对于Landau方程的Cauchy问题，在适当的初始数据下，包含一个尺度参数$a\in [\frac{2}{3},1]$，我们构造了唯一的全局时间一致的$ε$解，该解围绕一个局部Maxwell，其流体量是相应Euler系统的稀疏波.同时，我们建立了黎曼稀疏波解以$ε^{\frac{3}{5}-\frac{2}{5}a}|\ln ε|$为$ε\to 0$的速率一致地远离$t=0$的收敛性。该证明基于在尺度变换$(t,x)\to (ε^{-a}t,ε^{-a}x)$下结合[19]和[32]的精细能量方法。
英文摘要：
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 $ε>0$ is vanishing. Precisely, whenever $ε>0$ 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-$ε$ 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 $ε^{\frac{3}{5}-\frac{2}{5}a}|\ln ε|$ as $ε\to 0$. The proof is based on the refined energy approach combining [19] and [32] under the scaling transformation $(t,x)\to (ε^{-a}t,ε^{-a}x)$.
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