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Homogenization of Non-linear Scalar Conservation Laws

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Abstract

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 \({L_{\rm loc}^1}\) .

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  1. CEREMADE-UMR 7534, Université Paris-Dauphine, Place du maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Anne-Laure Dalibard

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  1. Anne-Laure Dalibard
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Correspondence to Anne-Laure Dalibard.

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Dalibard, AL. Homogenization of Non-linear Scalar Conservation Laws. Arch Rational Mech Anal 192, 117–164 (2009). https://doi.org/10.1007/s00205-008-0123-7

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