Homogenization of Non-linear Scalar Conservation Laws

We study the limit as ε → 0 of the entropy solutions of the equation $${\partial_t u^\varepsilon + {\rm div}_x \left[A \left(\frac{x}{\varepsilon},u^\varepsilon \right)\right] =0}$$ . We prove that the sequence uε two-scale converges toward 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 $${L_{\rm loc}^1}$$ .