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Homogenization of nonlinear scalar conservation laws

Dalibard, Anne-Laure (2009), Homogenization of nonlinear scalar conservation laws, Archive for Rational Mechanics and Analysis, 192, 1, p. 117-164. http://dx.doi.org/10.1007/s00205-008-0123-7

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00154678/en/
Date
2009
Journal name
Archive for Rational Mechanics and Analysis
Volume
192
Number
1
Publisher
Springer
Pages
117-164
Publication identifier
Metadata
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Author(s)
Dalibard, Anne-Laure
Abstract (EN)
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 $L^1_{\text{loc}}$.
Subjects / Keywords
Homogenization ; Scalar conservation law ; Kinetic formulation