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Uniform Stability of Oscillatory Shocks for KdV-Burgers Equation

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

Abstract

In this paper, we study the viscous-dispersive shock profile with infinite oscillations of the Korteweg-de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock wave, including the rate at which the local extrema converge to the left end state towards the left far field. Then, by exploiting the structural properties of the shock, we show the $L^2$ contraction property of the shock profile under arbitrarily large perturbations, up to a time-dependent shift. This result implies both time-asymptotic stability and uniform stability with respect to the viscosity and dispersion coefficients. This uniformity yields zero viscosity-dispersion limits.

Uniform Stability of Oscillatory Shocks for KdV-Burgers Equation

Abstract

In this paper, we study the viscous-dispersive shock profile with infinite oscillations of the Korteweg-de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock wave, including the rate at which the local extrema converge to the left end state towards the left far field. Then, by exploiting the structural properties of the shock, we show the contraction property of the shock profile under arbitrarily large perturbations, up to a time-dependent shift. This result implies both time-asymptotic stability and uniform stability with respect to the viscosity and dispersion coefficients. This uniformity yields zero viscosity-dispersion limits.
Paper Structure (33 sections, 13 theorems, 412 equations, 4 figures, 1 table)

This paper contains 33 sections, 13 theorems, 412 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Properties of Shock Profile and Main Ideas of Proofs
  4. Structural Properties of Viscous-Dispersive Shocks
  5. Idea for the Proof of Theorem \ref{['thm:shock']}
  6. Idea for the Proof of Theorem \ref{['thm:main']}
  7. Case 1: $\delta=0$.
  8. Case 2: $0<\delta s\le \frac{1}{4}$.
  9. Case 3: $\delta s>\frac{1}{4}$.
  10. Proof of Theorem \ref{['thm:shock']}: Shock Properties
  11. Upper Bound for the Rightmost Local Maximum
  12. Proof of \ref{['u0-upper']} in Theorem \ref{['thm:shock']}.
  13. Decay Rates of Oscillations
  14. Proof of \ref{['inc-decay']} in Theorem \ref{['thm:shock']}.
  15. Proof of Theorem \ref{['thm:main']}: Contraction Property
  16. ...and 18 more sections

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 non-monotonic viscous-dispersive shock of burgers which connects $u_-$ and $u_+$. Let $u_0$ be given initial data satisfying $\int_{\mathbb R} (u_0-\tilde{u})^2 dx<+\infty$ and let $u\in\mathcal{X}_T$ denote the solution of burgers with initial data $u_0$. Then, the

Figures (4)

  • Figure 1: The left panel displays the phase portrait of solutions to \ref{['t00']} with $\delta=10$, $\varepsilon=1$ and $u_{\pm}=\mp 1$ in the $(\tilde{u},\tilde{u}')$-plane. The blue dotted curves correspond to solitary wave solutions of KdVB equation with $\varepsilon=0$, while the black solid curve represents the heteroclinic orbit for $\varepsilon=1$ and $u_--u_+=2$. This orbit spirals from $\tilde{u}=u_-$ to $\tilde{u}=u_+$ and corresponds to the shock profile $\tilde{u}(\xi)$ shown in the right panel.
  • Figure 2: Illustration of the heteroclinic orbit in comparison with auxiliary curves on each interval of $\xi$ with monotonic $\tilde{u}$. (Left) The dotted curves denote parabolas, such as $y=\lambda(u_0-\tilde{u})(\tilde{u}+s)$ in \ref{['ineq_00']} for the rightmost monotonic interval, which lie above $\left|\tilde{u}'\right|$. (Right) To obtain a sharper estimate for $u_0$, we show \ref{['ineq_02']}. The slope of $(u_0-\tilde{u})^\frac{1}{2}(\tilde{u}+s)$ is infinite at $u_0$, yielding a more accurate approximation than $(u_0-\tilde{u})(\tilde{u}+s)$. In the other monotonic intervals, we use functions of the form $\lambda_i(\tilde{u}-u_i)^\frac{1}{2}(u_{i-1}-\tilde{u})^\frac{1}{2}$ to fit the heteroclinic orbit.
  • Figure 3: Illustration of $J_i$ for odd $i$ and the induction argument.
  • Figure 4: Illustration of $J_i$, when $i$ is odd. The corresponding picture for even $i$ is similar.

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • ...and 7 more