Nonlinear stability of time-periodic viscous shocks

TL;DR

This paper establishes the nonlinear stability of time-periodic viscous shocks in systems of conservation laws by analyzing the linearized operator's spectral properties and deriving sharp Green's function bounds.

Contribution

It introduces a novel approach using spatial dynamics and Floquet theory to handle time-dependent operators and embedded eigenvalues in stability analysis.

Findings

Proves linear and nonlinear stability of time-periodic Lax shocks under spectral stability conditions.

Develops contour integral representation for the evolution operator in time-periodic settings.

Provides sharp pointwise bounds on Green's functions for these shocks.

Abstract

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.