Nonlinear stability of viscous roll waves

TL;DR

This paper proves that spectral stability guarantees nonlinear stability for viscous roll waves in shallow water equations, overcoming challenges from nonconservative terms and incomplete parabolicity using Lagrangian coordinates and decay estimates.

Contribution

It extends stability results to nonconservative viscous roll waves, addressing key issues with nonconservative terms and incomplete parabolicity in the analysis.

Findings

Spectral stability implies nonlinear stability for viscous roll waves.

Lagrangian coordinates enable large-amplitude nonlinear damping estimates.

Nonconservative source terms decay faster than expected, aiding stability analysis.

Abstract

Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, which leads to undifferentiated quadratic source terms that cannot be handled using the estimates of the conservative case. The first is resolved by treating the equations in the more favorable Lagrangian coordinates, for which one can obtain large-amplitude nonlinear damping estimates similar to those carried out by Mascia--Zumbrun in the related shock wave case, assuming only symmetrizability of the hyperbolic part. The second is resolved by the observation that, similarly as in the relaxation and detonation cases, sources occurring in nonconservative components experience greater than expected decay, comparable to that experienced by a differentiated source.