Homogenization of nonlinear scalar conservation laws

TL;DR

This paper investigates the homogenization of nonlinear scalar conservation laws, showing that entropy solutions converge to a kinetic-type limit problem with mixed macroscopic and microscopic variables, and establishes strong convergence in local L1.

Contribution

It introduces a novel homogenization approach where the limit is a kinetic equation with mixed variables, diverging from traditional scalar conservation law limits.

Findings

Two-scale convergence of entropy solutions to a kinetic limit

Uniqueness of the limit kinetic equation

Strong local L1 convergence of solutions

Abstract

We study the limit as $\e\to 0$ of the entropy solutions of the equation $\p_t \ue + \dv_x[A(\frac{x}{\e},\ue)] =0$. We prove that the sequence $\ue$ two-scale converges towards a function $u(t,x,y)$, and $u$ is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in $L^1_{\text{loc}}$.