Homogenization of nonlinear scalar conservation laws
Résumé
We study the limit as $\e\to 0$ of the entropy solutions of the equation $\p_t \ue + \dv_x\left[A\left(\frac{x}{\e},\ue\right)\right] =0$. 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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