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Global existence and stability of near-affine solutions of compressible elastodynamics

Xianpeng Hu, Yuanzhi Tu, Changyou Wang, Huanyao Wen

TL;DR

This work proves global existence and nonlinear stability of near-affine solutions for the compressible elastodynamics system in $\mathbb{R}^d$ ($d=2,3$) under small $H^3$ perturbations. The authors reformulate the Cauchy problem in a Lagrangian frame around an affine flow, introduce perturbations, and develop a time-weighted energy method that leverages damping from the expanding affine map and diffusion from the elastic stress. A crucial a priori estimate for a Lyapunov functional $\mathcal{L}(t)$ closes via a bootstrap argument, yielding global strong solutions and explicit decay rates; the results are then transported to Eulerian coordinates using the flow map, with detailed regularity in $L^2$-based spaces. The paper thus establishes nonlinear stability of the non-constant affine solution in compressible elastodynamics and provides a rigorous bridge between Lagrangian and Eulerian descriptions with quantified decay for density, velocity, and deformation gradient.

Abstract

We prove that for sufficiently small $H^3$-perturbations of an affine solution, the Cauchy problem for the compressible nonlinear elastodynamics in $\mathbb{R}^d$, for $d=2,3$, admits a unique global strong solution. Moreover, we establish the asymptotic behavior of the solution.

Global existence and stability of near-affine solutions of compressible elastodynamics

TL;DR

This work proves global existence and nonlinear stability of near-affine solutions for the compressible elastodynamics system in () under small perturbations. The authors reformulate the Cauchy problem in a Lagrangian frame around an affine flow, introduce perturbations, and develop a time-weighted energy method that leverages damping from the expanding affine map and diffusion from the elastic stress. A crucial a priori estimate for a Lyapunov functional closes via a bootstrap argument, yielding global strong solutions and explicit decay rates; the results are then transported to Eulerian coordinates using the flow map, with detailed regularity in -based spaces. The paper thus establishes nonlinear stability of the non-constant affine solution in compressible elastodynamics and provides a rigorous bridge between Lagrangian and Eulerian descriptions with quantified decay for density, velocity, and deformation gradient.

Abstract

We prove that for sufficiently small -perturbations of an affine solution, the Cauchy problem for the compressible nonlinear elastodynamics in , for , admits a unique global strong solution. Moreover, we establish the asymptotic behavior of the solution.

Paper Structure

This paper contains 10 sections, 7 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Global well-posedness in the Lagrangian coordinate
  3. Reformulation in the Lagrangian coordinate
  4. Perturbations of affine solutions
  5. Local existence and uniqueness
  6. A priori estimates
  7. Proof of Proposition \ref{['uniform_estimates']}
  8. Proof of Theorem \ref{['global_existence_Lagrangian']}
  9. Global well-posedness in Eulerian coordinate
  10. Regularity estimates

Key Result

Theorem 1.1

For $d=2, 3$, there exists a diffeomorphism $\xi_0: \mathbb{R}^d \to \mathbb{R}^d$ and a small constant $\varepsilon>0$ such that if the initial data $(\rho_0, u_0, F_0)$, with $(\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\in H^3_x(\mathbb{R}^d)$, satisfies then the Cauchy problem elasticity_equation_1 admits a unique global strong solution $(\rho, u, F)$ such that $(\rho(x,t) -\frac{1}{(1+t)^d}, u(x,t)-

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.1
  • ...and 1 more