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Local energy decay of solutions to the linearized compressible viscoelastic system around motionless state in an exterior domain

Yusuke Ishigaki, Takayuki Kobayashi

TL;DR

This work analyzes the large-time behavior of perturbations for the linearized compressible viscoelastic system around a motionless state in a three-dimensional exterior domain. By conducting a resolvent analysis that combines whole-space and bounded-domain problems via a parametrix, the authors establish a local energy decay of the linearized semigroup with rate $t^{-2-m}$ for time-derivative orders $m$, capturing diffusion from viscous effects and wave propagation from elastic shear and sound modes. A key finding is the low-frequency singularity in the resolvent of the form $\lambda^2\log\lambda$ (distinct from the β=0 case where the singularity is weaker), which underpins the observed diffusion-wave decay and the faster local energy decay relative to classical heat-like behavior. The results extend prior β=0 exterior-domain analyses to the coupled viscoelastic system, providing precise mapping properties of the resolvent operators and a robust analytic framework for the linear evolution in exterior domains.

Abstract

We study the large time behavior of solutions to the system of equations describing motion of compressible viscoelastic fluids. We focus on the linearized system around a motionless state in a three-dimensional exterior domain and derive the local energy decay estimate of its solution to give the diffusion wave phenomena caused by sound wave viscous diffusion and elastic shear wave.

Local energy decay of solutions to the linearized compressible viscoelastic system around motionless state in an exterior domain

TL;DR

This work analyzes the large-time behavior of perturbations for the linearized compressible viscoelastic system around a motionless state in a three-dimensional exterior domain. By conducting a resolvent analysis that combines whole-space and bounded-domain problems via a parametrix, the authors establish a local energy decay of the linearized semigroup with rate for time-derivative orders , capturing diffusion from viscous effects and wave propagation from elastic shear and sound modes. A key finding is the low-frequency singularity in the resolvent of the form (distinct from the β=0 case where the singularity is weaker), which underpins the observed diffusion-wave decay and the faster local energy decay relative to classical heat-like behavior. The results extend prior β=0 exterior-domain analyses to the coupled viscoelastic system, providing precise mapping properties of the resolvent operators and a robust analytic framework for the linear evolution in exterior domains.

Abstract

We study the large time behavior of solutions to the system of equations describing motion of compressible viscoelastic fluids. We focus on the linearized system around a motionless state in a three-dimensional exterior domain and derive the local energy decay estimate of its solution to give the diffusion wave phenomena caused by sound wave viscous diffusion and elastic shear wave.

Paper Structure

This paper contains 10 sections, 19 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Problems
  3. Known results
  4. Main results
  5. Outline of Theorem \ref{['Local-energy']}
  6. Notation
  7. Resolvent problem in the whole space
  8. Zero-resolvent problem in the bounded domain
  9. Resolvent problem in the exterior domain
  10. Proof of Theorem \ref{['Local-energy']}

Key Result

Theorem 1.1

Let $1<p<\infty$, let $b_0>0$ be $\mathbb{R}^3\setminus\Omega\subset B_{b_0}$ and let $b>b_0$. If $u_0=(\phi_0,w_0,G_0)$ satisfies $u_0\in W^{1,p}(\Omega)\times L^p(\Omega)^3\times W^{1,p}(\Omega)^{3\times3}$, $\nabla\phi_0+\mathrm{div}{}^\top G_0=0$, $G_0\in \nabla(W^{2,p}(\Omega)\cap W_0^{1,p}(\Om where $C_{b,b_0,m,p}$ is a constant depending only on $b,b_0,m$ and $p$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof : Proof of Proposition \ref{['R-R3-prop1']}
  • Proposition 3.4
  • Corollary 3.5
  • Remark 3.6
  • ...and 20 more