Regularizing effect for conservation laws with a Lipschitz convex flux

Billel Guelmame 1, 2 Stéphane Junca 1, 2 Didier Clamond 1
2 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
Abstract : This paper studies the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no additional regularity is assumed on the flux. Thus, we generalize the well-known $\mathrm{BV}$ smoothing effect for $\mathrm{C}^2$ uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik, while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be dicontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. We modify the wave velocity in order to get an Oleinik inequality useful for the wave front tracking algorithm. Then, we prove that the unique entropy solution belongs to a generalized $\mathrm{BV}$ space, $\mathrm{BV}^\Phi$.
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https://hal.archives-ouvertes.fr/hal-01943834
Contributor : Stéphane Junca <>
Submitted on : Tuesday, December 4, 2018 - 10:56:07 AM
Last modification on : Tuesday, October 29, 2019 - 12:08:21 PM

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Billel Guelmame, Stéphane Junca, Didier Clamond. Regularizing effect for conservation laws with a Lipschitz convex flux. Communications in Mathematical Sciences, International Press, In press. ⟨hal-01943834⟩

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