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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.

Relative entropy and contraction for extremal shocks of conservation laws up to a shift