Homogenization of nonlinear scalar conservation laws

Details Homogenization of nonlinear scalar conservation laws Homogenization of nonlinear scalar conservation laws

2007-06-14

arxiv.org/abs/0706.2104v1

Archive for Rational Mechanics and Analysis (2008) ISSN: 0003-9527 (Print) 1432-0673 (Online)

Mathematics

Analysis of PDEs

34 pages

Scientific paper

10.1007/s00205-008-0123-7

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 $L^1_{\text{loc}}$.

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