Remarks on Nonlinear Elastic Waves in the Radial Symmetry in 2-D

TL;DR

This paper establishes a variational framework for nonlinear elastic waves in 2-D isotropic materials, introduces a null condition to prevent singularities, and proves global existence of smooth solutions under radial symmetry with small initial data.

Contribution

It provides the first explicit variational structure for 2-D nonlinear elastic waves and demonstrates how a null condition ensures global solutions in the radial case.

Findings

Null condition prevents finite-time singularities.

Global existence of smooth solutions with small initial data.

Explicit variational structure for 2-D nonlinear elastic waves.

Abstract

In this manuscript we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind of structural condition on the nonlinearity in order to stop the formation of finite time singularities of local smooth solutions. In the radial symmetric case, inspired by Alinhac's work [3] on 2-D quasilinear wave equations, we show that such null condition can ensure the globalexistence of smooth solutions with small initial data.