Stability of Viscous Profiles: Proofs Via Dichotomies

In this paper we give a self-contained approach to a nonlinear stability result, as t → ∞, for a viscous profile corresponding to a strong shock of a system of conservation laws. The initial perturbation is assumed to be small and to have zero mass. As t→ ∞, the solution with perturbed initial data is shown to approach the viscous profile in maximum norm.A complete proof of the stability result is given under slightly weaker assumptions than those in Comm. Pure Appl. Math. LI (1998) 1397]; our assumptions, techniques, and results also differ from those in Indiana Univ. Math. J. 47 (1998) 741]. To derive resolvent estimates for a linearized problem, we use the theory of exponential dichotomies for ODEs extensively. A main tool provided by this theory is a quantitative L ₁ perturbation theorem for dichotomies, which yields the delicate resolvent estimates for s near zero.When showing that the resolvent estimates imply nonlinear stability, we essentially follow the arguments in Comm. Pure Appl. Math. LI (1998) 1397; SIAM J. Math. Anal. 20 (1999) 401], but note some simplifications.