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In this paper, we study the [Formula: see text]-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient [Formula: see text] can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for [Formula: see text] conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for [Formula: see text] systems of hyperbolic conservation laws are proved.

Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of smallBVfunctions which are global solutions of this equation. For any smallBVinitial data, such global solutions are known to exist. Moreover, they are known to be unique amongBVsolutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not evenBV) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-BVuniqueness result, where only one of the solutions is supposed to be smallBV, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in theBVtheory, in the case of systems with 2 unknowns. The method is$$L^2$$$L^2$based, and builds up from the theory of a-contraction with shifts, where suitable weight functionsaare generated via the front tracking method.