Invariant Manifolds and the Stability of Traveling Waves in Scalar Viscous Conservation Laws ∗

The stability of traveling wave solutions of scalar, viscous conservation laws is inves-tigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field compo-nents are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same tem-poral decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces. ∗Key words: invariant manifolds, stability of traveling waves, scalar viscous conservation laws, weighted function spaces