Uniform Stability of Oscillatory Shocks for KdV-Burgers Equation

TL;DR

This paper analyzes the stability of oscillatory shock waves in the KdV-Burgers equation, establishing detailed structures and demonstrating uniform stability and stability in the zero viscosity-dispersion limit.

Contribution

It provides a detailed structural analysis of viscous-dispersive shocks and proves their uniform stability under large perturbations, including stability in the zero viscosity-dispersion limit.

Findings

Shock waves have detailed convergence rates to far field states.

Shock profiles exhibit $L^2$-contraction under large perturbations.

Zero viscosity-dispersion limits lead to orbitally stable Riemann shocks.

Abstract

We study viscous-dispersive shock waves with infinite oscillations of the Korteweg-de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock waves, including the rates at which the local extrema converge to the left end state towards the left far field. Then, by exploiting the structural properties of the shocks, we show the $L^2$-contraction property of the shock profiles under arbitrarily large perturbations, up to time-dependent shifts. This property implies both time-asymptotic stability and uniform stability with respect to the viscosity and dispersion coefficients. This uniformity yields the existence of zero viscosity-dispersion limits, on which Riemann shocks are orbitally stable.