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Regularizing effect for conservation laws with a Lipschitz convex flux. (English) Zbl 1439.35329

Summary: 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 \(\operatorname{BV}\) smoothing effect for \(\mathrm{C}^2\) uniformly convex fluxes discovered independently by P. D. Lax [Commun. Pure Appl. Math. 10, 537–566 (1957; Zbl 0081.08803)] and O. A. Oleĭnik [Transl., Ser. 2, Am. Math. Soc. 26, 95–172 (1963; Zbl 0131.31803)], while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be discontinuous 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 \(\operatorname{BV}\) space, \(\operatorname{BV}^\Phi\).

MSC:

35L67 Shocks and singularities for hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
35L65 Hyperbolic conservation laws
26A45 Functions of bounded variation, generalizations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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