Fractional BV spaces and applications to scalar conservation laws

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BV s (ℝ) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space W s,1/s (ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BV s spaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BV s initial data. Furthermore, for the first time, we get the maximal W s,p smoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.