Center stable manifolds for quasilinear parabolic pde and conditional stability of nonclassical viscous shock waves
Abstract
Description
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. As an application, we show conditional stability of Lax- or undercompressive shock waves of general quasilinear parabolic systems of conservation laws by a pointwise stability analysis on the center stable manifold.