Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$L^2$-contraction of Shock Waves for KdV-Burgers Equation

Geng Chen, Namhyun Eun, Moon-Jin Kang, Yannan Shen

Abstract

The KdV-Burgers equation is a canonical model describing the interplay between nonlinearity, viscosity and dispersion, and it admits viscous-dispersive shocks as traveling wave solutions. In this paper, we establish an $L^2$-contraction property for viscous-dispersive shocks under arbitrarily large perturbations, up to a time-dependent shift. This yields time-asymptotic stability and uniform estimates with respect to the strengths of viscosity and dispersion. We present the proof for the monotone shocks, and introduce the companion work in [6] on the stability and structural properties of oscillatory shocks.

$L^2$-contraction of Shock Waves for KdV-Burgers Equation

Abstract

The KdV-Burgers equation is a canonical model describing the interplay between nonlinearity, viscosity and dispersion, and it admits viscous-dispersive shocks as traveling wave solutions. In this paper, we establish an -contraction property for viscous-dispersive shocks under arbitrarily large perturbations, up to a time-dependent shift. This yields time-asymptotic stability and uniform estimates with respect to the strengths of viscosity and dispersion. We present the proof for the monotone shocks, and introduce the companion work in [6] on the stability and structural properties of oscillatory shocks.
Paper Structure (6 sections, 5 theorems, 60 equations)

This paper contains 6 sections, 5 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:main']}: Contraction Property
  5. Remark on Oscillatory Shocks
  6. Proof of Theorem \ref{['thm:exp']}: Exponential Decay

Key Result

Theorem 1.1

Let $\varepsilon,\delta>0$ be constants and let $u_\pm$ be given states. Assume that $u_->u_+$ and Let $\tilde{u}$ be the associated viscous-dispersive shock of burgers which monotonically connects $u_-$ and $u_+$. Let $u_0$ be given initial data with $\left\|u_0-\tilde{u}\right\|_{H^1({\mathbb R})}<+\infty$ and let $u\in\mathcal{X}_T$ denote the solution of burgers subject to the initial data $u

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3