Regularizing effect for conservation laws with a Lipschitz convex flux

Abstract: This talk deals with  the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on Real line. Beside convexity, no additional regularity is assumed on the flux. Thus, the well-known   BV smoothing effect for   C²  uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik  is generalized for fluxes 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.  The wave velocity is modified in order to get an Oleinik inequality useful for a new modified wave front tracking algorithm. The  unique entropy solution can not be a function with bounded variation but belongs to a generalized BV space which depends only on the non-linearity of the flux.