Smoothing effect in BVΦ for entropy solutions of scalar conservation laws

This paper deals with a sharp smoothing e ect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex ux. We brie y explain why degenerate uxes are related with the optimal smoothing e ect 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 spacesto quantify the regularizing e ect and to obtain traces as in BV. The function Φ in question is linked to the degeneracy of the ux. Up to the present, the Lax-Ole nik formula has provided optimal results for a uniformly convex ux. This formula is validated in this paper for the more general class of C1 strictly convex uxes -which contains degenerate convex uxesand enables the BVΦ smoothing e ect in this class. We give a complete proof that for a C 1 strictly convex ux the Lax-Ole nik formula provides the unique entropy solution, namely the Kruokov solution.