Fractional BV spaces for conservation laws

Abstract:  In 1994, a Lions-Perthame-Tadmor conjecture proposes an optimal regularity in fractional Sobolev spaces for entropy solutions of nonlinear multidimensional scalar conservation laws.   The first optimal result for uniformly convex fluxes by Lax and Oleinik 1957 is stated in BV (space of functions with bounded variation) which gives the sharp Sobolev regularity and, moreover, the trace property on each side of a shock wave. Unfortunately "BV is not enough".  Fractional BV spaces and nonlinear sets of rescaled functions in BV are proposed in this talk  in order to preserve the "BV trace property" and the optimal  Sobolev exponent.  Recent optimal results revisiting the one-sided Oleinik inequality, reformulations of the Lions-Perthame-Tadmor  conjecture  and the total oscillation diminishing property given by  Perthame and Westdickenberg in 2005 are presented in this framework.