Stability of viscous profiles: Proofs via dichotomies
In this paper we give a self-contained approach to a nonlinear stability result, as t ->infinity, 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 ->infinity, 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 App/. Math. L1 (1998) 1397; SIAM J. Math. Anal. 20 (1999) 401], but note some simplifications.
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Springer US