Uniqueness and Weak-BV Stability for $$2\times 2$$ Conservation Laws
Geng Chen, Sam G. Krupa, Alexis Vasseur
Abstract
Abstract Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small BV functions which are global solutions of this equation. For any small BV initial data, such global solutions are known to exist. Moreover, they are known to be unique among BV solutions 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 even BV ) 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- BV uniqueness result, where only one of the solutions is supposed to be small BV , 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 the BV theory, in the case of systems with 2 unknowns. The method is $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> based, and builds up from the theory of a-contraction with shifts, where suitable weight functions a are generated via the front tracking method.