Nonlinear stability of rarefaction waves for systems of viscous hyperbolic conservation laws.

Abstract

We study the asymptotic convergence to rarefaction waves of the solution for the initial value problem for some systems of hyperbolic conservation laws with positive viscosity. First, we prove that for 2 x 2, strictly hyperbolic and strongly coupled system, if the k$\sp{th}$ characteristic field is genuinely nonlinear, then a weak k-rarefaction wave is nonlinearly stable in the sense that if the initial data is close to the initial data for a weak k-rarefaction wave for corresponding hyperbolic conservation laws, then the solution tends to the rarefaction wave as t tends to infinity. Next, we prove that in addition to the above assumptions, if the system is genuinely nonlinear, then a linear superposition of weak rarefaction waves from both characteristic families is nonlinearly stable. Finally, we get similar results for Euler equations with artificial viscosity. The proofs of our results consist mainly of a priori estimates on the perturbations of smooth rarefaction waves constructed by making use of the inviscid Burger's equation. These estimates are proved by using elementary energy methods, weighted characteristic-energy methods and their combinations, all of which are based on the fact that a smooth rarefaction wave is expansive and its derivatives decay in time algebraically.