  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 BVs spaces wider than BV are required, where
the exponent 0<s\leq 1 and BV = BV1. 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 BVs 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.
