Scattering and Rigidity for Nonlinear Elastic Waves

TL;DR

This paper studies the long-term behavior of solutions to nonlinear elastic wave equations, proving they scatter to linear solutions and establishing a rigidity result linking vanishing scattering data to trivial solutions.

Contribution

It demonstrates the scattering behavior and rigidity property for global solutions of 3D nonlinear elastic waves satisfying the null condition, utilizing the system's variational structure.

Findings

Global solutions scatter to linear elastic waves as time approaches infinity.

Vanishing scattering data implies the solution is identically zero.

The variational structure is crucial in proving these results.

Abstract

For the Cauchy problem of nonlinear elastic wave equations of three dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225--250) and T. C. Sideris (Ann. Math. 151(2000) 849--874), independently. In this paper, we will consider the asymptotic behavior of global solutions. We first show that the global solution will scatter, i.e., it will converge to some solution of linear elastic wave equations as time tends to infinity, in the energy sense. We also prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically. The variational structure of the system will play a key role in our argument.