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Stability of Rarefaction Waves Under Periodic Perturbation for A Rate-Type Viscoelastic System

lin Chang, Duo Liu, Weiqiang Zhang

Abstract

In this paper, a rarefaction wave under space-periodic perturbation for the 3 times 3 rate-type viscoelastic system is considered. It is shown that if the initial perturbation around the rarefaction wave is suitably small, then the solution of the rate-type viscoelastic system tends to the rarefaction wave. The stability of solutions under periodic perturbations is an interesting and important problem since the perturbation keeps oscillating at the far fields. That is, the perturbation is not integral in space. The key of proof is to construct a suitable ansatz carrying the same oscillation as the solution. Then we can find cancellations between solutions and ansatz such that the perturbation belongs to some Sobolev space. The nonlinear stability can be obtained by the weighted energy method.

Stability of Rarefaction Waves Under Periodic Perturbation for A Rate-Type Viscoelastic System

Abstract

In this paper, a rarefaction wave under space-periodic perturbation for the 3 times 3 rate-type viscoelastic system is considered. It is shown that if the initial perturbation around the rarefaction wave is suitably small, then the solution of the rate-type viscoelastic system tends to the rarefaction wave. The stability of solutions under periodic perturbations is an interesting and important problem since the perturbation keeps oscillating at the far fields. That is, the perturbation is not integral in space. The key of proof is to construct a suitable ansatz carrying the same oscillation as the solution. Then we can find cancellations between solutions and ansatz such that the perturbation belongs to some Sobolev space. The nonlinear stability can be obtained by the weighted energy method.
Paper Structure (9 sections, 9 theorems, 82 equations)

This paper contains 9 sections, 9 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Suitable Ansatz
  4. Main Theorem
  5. Reformulation of the problem
  6. A priori estimate
  7. The proof of theorem \ref{['theorem201']}
  8. Appendix
  9. Proof of Lemma \ref{['Lem-H']}

Key Result

Lemma 2.1

(HP1999) Under the condition $(2.4)$, there exists a smooth function $(V^r(x,t),U^r(x,t))$, which is the smooth approximation of $(u^r, u^r)$ satisfying the following properties

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.1
  • Proposition 3.1: A priori estimate
  • Lemma 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 3 more