We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space . Since C. de Lellis and M. Westdickenberg  have proved the existence of such solutions that do not belong to if either or and or with and , this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.
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François Golse, Benoît Perthame, Optimal regularizing effect for scalar conservation laws. Rev. Mat. Iberoam. 29 (2013), no. 4, pp. 1477–1504DOI 10.4171/RMI/765