Viscous Destabilization for Large Shocks of Conservation Laws

Abstract. The recent theory of [Formula: see text]contraction with shifts provides [Formula: see text]-stability for shock waves of one-dimensional hyperbolic systems of conservation laws. The theory has been established at the inviscid level uniformly in the shock amplitude and at the viscous level for small shocks. In this work, we investigate whether the [Formula: see text]contraction property holds uniformly in the shock amplitude for some specific systems with viscosity. We show that in some cases, the [Formula: see text]contraction fails for sufficiently large shocks. This showcases a “viscous destabilization” effect in the sense that the [Formula: see text]-contraction property is verified for the inviscid model but can fail for the viscous one. This also shows that the [Formula: see text]-contraction property, even among small perturbations, is stronger than the classical notion of nonlinear stability, which is known to hold regardless of shock amplitude for viscous scalar conservation laws.