Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux

Moon-Jin Kang, HyeonSeop Oh

TL;DR

This work addresses quantitative stability and decay for large perturbations of planar viscous shocks in multi-D scalar viscous conservation laws with strictly convex flux. It develops an $a$-contraction with shifts framework in the setting $\Omega = \mathbb{R} \times \mathbb{T}^2$, constructing a weight function and a dynamical shift to obtain contraction of the relative entropy and, under an $L^1$-integrable perturbation, a decay rate of $t^{-1/4}$ in $L^2$ toward the planar shock. The analysis hinges on a two-region decomposition of perturbations, Taylor expansions near the shock, and a novel multi-dimensional nonlinear Poincaré-type inequality with constraints to control cubic nonlinearities, together with precise truncation and comparison arguments. The results yield a quantitative description of convergence to planar shocks under large perturbations and establish uniform bounds on the shift, providing a significant step in the understanding of multi-D shock stability for viscous conservation laws with a single entropy.

Abstract

We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in $L^2$ of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial $L^2$-perturbation is also in $L^1$, then the large perturbation decays of rate $t^{-1/4}$ in $L^2$, up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations.

Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux

TL;DR

This work addresses quantitative stability and decay for large perturbations of planar viscous shocks in multi-D scalar viscous conservation laws with strictly convex flux. It develops an -contraction with shifts framework in the setting , constructing a weight function and a dynamical shift to obtain contraction of the relative entropy and, under an -integrable perturbation, a decay rate of in toward the planar shock. The analysis hinges on a two-region decomposition of perturbations, Taylor expansions near the shock, and a novel multi-dimensional nonlinear Poincaré-type inequality with constraints to control cubic nonlinearities, together with precise truncation and comparison arguments. The results yield a quantitative description of convergence to planar shocks under large perturbations and establish uniform bounds on the shift, providing a significant step in the understanding of multi-D shock stability for viscous conservation laws with a single entropy.

Abstract

We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial -perturbation is also in , then the large perturbation decays of rate in , up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations.
Paper Structure (21 sections, 19 theorems, 224 equations)

This paper contains 21 sections, 19 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Main ideas of proof
  4. Organization of paper
  5. Preliminaries
  6. Properties of weak shock
  7. Relative entropy method
  8. Construction of the weight function
  9. Global estimates on the relative entropy
  10. Multi-dimensional nonlinear Poincaré type inequality with constraints
  11. Proof of Theorem \ref{['main']}
  12. Definition of the shift function
  13. Proof of Theorem \ref{['main']} from a main proposition
  14. Proof of Proposition \ref{['mainprop']}
  15. Expansion in the size of the shock
  16. ...and 6 more sections

Key Result

Theorem 1.1

Consider SVCL with a strictly convex flux $f_1$ satisfying the growth condition growth. Let $\eta$ be an entropy which satisfies (A1)-(A2). For any $u_- \in \mathbb{R}$, there exists $\delta_0 \in (0,1)$ such that the following holds. For any $\varepsilon, \lambda >0$ with $\delta_0^{-1} \varepsilon Here, the shift function $X(t)$ satisfies $X \in W_{loc}^{1,1}((0,T))$ and for some positive funct

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.1
  • ...and 19 more