Fractional BV spaces and first applications to scalar conservation laws

TL;DR

This paper explores fractional BV spaces and their application to scalar conservation laws, establishing stability and smoothing effects for entropy solutions with less regular initial data.

Contribution

It introduces the use of fractional BV spaces in scalar conservation laws, proving stability and maximal smoothing effects for entropy solutions with minimal regularity.

Findings

Established stability of entropy solutions with fractional BV initial data.

Proved maximal W^{s,p} smoothing effect for nonlinear degenerate convex fluxes.

Connected fractional BV spaces to critical Sobolev spaces in this context.

Abstract

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.