Remarks on the stability of shock profiles for conservation laws with dissipation

Two remarks are made. The first is to establish the stability of monotone shock profiles of the KdV-Burgers equation, based on an energy method of Goodman. The second remark illustrates, specifically in Burgers' equation, that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms. Introduction. We will prove two results. The first is to establish the orbital stability of monotone shock profile solutions 4(x st) of the KdV-Burgers equation (1.1) ut + uux uxx-uxxx =0, ? > 0. The second is to show that for shock profile solutions of viscous conservation laws, and specifically for Burgers' equation (1.2) ut + uux -uxx = ? one cannot expect perturbations to decay at any uniform rate (algebraic or otherwise) in ordinary norms, such as LP. The first result is related to Goodman's conditional stability result in [5] for small amplitude shock profiles +(x st) in systems of genuinely nonlinear conservation laws with viscosity, u, + f(u)x = Buxx u e Rm. The second is intended to be relevant to the question of full stability for shock profiles of such systems. The nonlinear stability to small perturbations of constant state solutions u0(x) c of the system u, + f(u)x = (B(u)ux)x, where B(c) is a strictly stable viscosity matrix (see [9]), can be demonstrated based on a uniform rate of decay of solutions to the linearized equation. The method used is similar to that developed in [8] and [10] for systems in n > 1 space dimensions. Faster decay rates in higher space dimensions make the method easier to apply when n is large, but it does just work for conservation laws with stable viscosities in one space dimension (see [13]). Thus, for example, assuming c = 0, if IIuo IIH6 LL is sufficiently small, then IIu(*, t)llH4 < Ct/1/4IIUoII H4 n LA In this note, we want to point out that such uniform decay rates cannot be valid for perturbations of shock profiles. In particular, for the shock profile solution of Burgers' equations we construct a family of perturbations which decay to zero but do so at no uniform rate in most ordinary norms (including all LP norms). Thus, for Received by the editors October 15, 1984 and, in revised form, December 19, 1984. 1980 Mathematics Subject Classification. Primary 35Q20, 35L65. ?1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page