Asymptotic stability of traveling waves for viscous conservation laws with dispersion

Abstract

This work is concerned with the asymptotic stability of traveling waves for scalar viscous conservation laws with a convex flux function and a dispersion term. First we prove the existence of solutions locally in time of the initial-value problem for initial data near a constant solution by Fourier analysis. Using the semigroup method the local existence for initial data that are an $L^2$ perturbation of a traveling-wave profile is proved. We also obtain a regularity property of these solutions. The solution operator generated by the linearized equation plays a crucial role. Using the energy method we establish a priori estimates. These estimates, when combined with the local existence, lead to the desired global-in-time existence as well as the time-asymptotic decay of solutions with initial data close to a monotone traveling wave.