Homogenization of nonlinear scalar conservation laws

We study the limit as $\text{e}\to 0$ of the entropy solutions of the equation ${\text{p}}_t\text{ue}+{\text{dv}}_x[A(\frac{x}{\text{e}},\text{ue})]=0$. We prove that the sequence $\text{ue}$ 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 $L_{\text{loc}}^1$.