Guelmame, Billel; Junca, Stéphane; Clamond, Didier Regularizing effect for conservation laws with a Lipschitz convex flux. (English) Zbl 1439.35329 Commun. Math. Sci. 17, No. 8, 2223-2238 (2019). 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, \(\operatorname{BV}^\Phi\). Cited in 4 Documents 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.) Keywords:scalar conservation laws; entropy solution; strictly convex flux; discontinuous velocity; wave front tracking; smoothing effect; \(\operatorname{BV}^\Phi\) spaces Citations:Zbl 0081.08803; Zbl 0131.31803 PDFBibTeX XMLCite \textit{B. Guelmame} et al., Commun. Math. Sci. 17, No. 8, 2223--2238 (2019; Zbl 1439.35329) Full Text: DOI