Regularizing effect for conservation laws with a Lipschitz convex flux

Effet régularisant pour des lois de conservation avec flux Lipschitz strictement convexe

Billel Guelmame (1, 2) , Stéphane Junca (1, 2) , Didier Clamond (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 COFFEE - COmplex Flows For Energy and Environment

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, $\mathrm{BV}^\Phi$.

Mots clés: $\mathrm{BV}^\Phi$ spaces; Scalar conservation laws; discontinuous velocity; smoothing effect; wave front tracking; entropy solution; strictly convex flux

Domaines: Equations aux dérivées partielles [math.AP]

Billel Guelmame, Stéphane Junca, Didier Clamond. Regularizing effect for conservation laws with a Lipschitz convex flux. Communications in Mathematical Sciences, 2019, 17 (8), pp.2223-2238. ⟨10.4310/CMS.2019.v17.n8.a6⟩. ⟨hal-01943834⟩