Relative entropy and contraction for extremal shocks of Conservation Laws up to a shift

Abstract: We consider systems of conservation laws endowed with a convex entropy. We show the contraction, up to a translation, to extremal entropic shocks, for a pseudo-distance based on the notion of relative entropy. The contraction holds for bounded entropic weak solutions having an additional trace property. The pseudo-distance depends only on the fixed extremal entropic shocks in play. In particular, it can be chosen uniformly for any entropic weak solutions which are compared to a fixed shock. The boundedness of the solutions controls the strength of the shift needed to get the contraction. However, no $BV$ estimate is needed on the weak solutions considered. The theory holds without smallness condition. For fluid mechanics, the theory handles solutions with vacuum.