Smoothing effect in BVΦ for entropy solutions of scalar conservation laws

Abstract

This paper deals with a sharp smoothing effect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex flux. We briefly explain why degenerate fluxes are related with the optimal smoothing effect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional conservation laws. It turns out that generalized spaces of bounded variation BVΦ are particularly suitable –better than Sobolev spaces– to quantify the regularizing effect and to obtain traces as in BV. The function Φ in question is linked to the degeneracy of the flux. Up to the present, the Lax–Oleĭnik formula has provided optimal results for a uniformly convex flux. This formula is validated in this paper for the more general class of C1 strictly convex fluxes –which contains degenerate convex fluxes– and enables the BVΦ smoothing effect in this class. We give a complete proof that for a C1 strictly convex flux the Lax–Oleĭnik formula provides the unique entropy solution, namely the Kruzhkov solution.