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We develop a theory based on relative entropy to show the uniqueness and L^2stability (up to a translation) of extremal entropic Rankine-Hugoniotdiscontinuities for systems of conservation laws (typically 1-shocks, n-shocks,1-contact discontinuities and n-contact discontinuities of large amplitude)among bounded entropic weak solutions having an additional trace property. Theexistence of a convex entropy is needed. No BV estimate is needed on the weaksolutions considered. The theory holds without smallness condition. Theassumptions are quite general. For instance, strict hyperbolicity is not neededglobally. For fluid mechanics, the theory handles solutions with vacuum.

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations