Fractional regularity for conservation laws with discontinuous flux

Overview & Synopsis

This article deals with the regularity of the entropy solutions of scalar conservation laws with discontinuous flux. It is well-known [Adimurthi et al., Comm. Pure Appl. Math. 2011] that the entropy solution for such equation does not admit BV regularity in general, even when the initial data belongs to $BV$. Due to this phenomenon fractional $BV^s$ spaces wider than $BV$ are required, where the exponent $0 < s ≤ 1$ and $BV = BV^1$. It is a long standing open question to find the optimal regularizing effect for the discontinuous flux with $L^\infty$ initial data. The optimal regularizing effect in $BV^s$ is proven on an important case using control theory.The fractional exponent s is at most $1/2 $ even when the fluxes are uniformly convex.