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$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation

Moon-Jin Kang, HyeonSeop Oh

Abstract

We consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front. However, for the transversal directions, we do not have any restrictions on flux function. We first show the contraction property for any large perturbations in $L^2$ of the planar viscous shock. If the initial $L^2$-perturbation is also in $L^1$, the large perturbation converges to zero in $L^2$ as time goes to infinity with $t^{-1/4}$ decay rate. The contraction and decay estimates hold up to dynamical shift. For the results, we do not impose any smallness conditions on the initial value. This result extends the 1D case \cite{Kang-V-1} by the first author and Vasseur to the multi-dimensional case.

$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation

Abstract

We consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front. However, for the transversal directions, we do not have any restrictions on flux function. We first show the contraction property for any large perturbations in of the planar viscous shock. If the initial -perturbation is also in , the large perturbation converges to zero in as time goes to infinity with decay rate. The contraction and decay estimates hold up to dynamical shift. For the results, we do not impose any smallness conditions on the initial value. This result extends the 1D case \cite{Kang-V-1} by the first author and Vasseur to the multi-dimensional case.
Paper Structure (3 sections, 4 theorems, 60 equations)

This paper contains 3 sections, 4 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Proof of contraction estimate
  3. Proof of decay estimate

Key Result

Theorem 1.1

Consider VSCL with Bflux. Given two constants $u_-$ and $\varepsilon$ with $0<\varepsilon < 8\pi (2a+\|g" \|_{L^{\infty}(\mathbb R)})^{-1}$, let $u_+ = u_- - \varepsilon$. Let ${\tilde{u}}$ be the viscous shock wave as in Vshock. Then, for any solution $u$ of VSCL with initial data $u_0$ satisfying moreover, for all $t>0$, and

Theorems & Definitions (5)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.1