Fractional BV spaces and first applications to scalar conservation laws

The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some "fractional $BV$ spaces" denoted $BV^s$, for $0 < s \leq 1$, introduced by Love and Young in 1937. The $BV^s(\R)$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\R)$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow 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 P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes.