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

TL;DR

This paper proves the global existence and stability of solutions close to affine solutions in compressible elastodynamics, demonstrating that small perturbations lead to unique, long-term solutions with predictable asymptotic behavior.

Contribution

It establishes the global existence and stability of near-affine solutions for compressible elastodynamics in 2D and 3D, a significant extension in understanding their long-term behavior.

Findings

Unique global strong solutions exist for small perturbations.

Solutions exhibit predictable asymptotic decay.

Stability holds in the Sobolev space $H^3$.

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.