The null condition in elastodynamics leads to non-uniqueness

TL;DR

This paper demonstrates that in two-dimensional elastodynamics with a null condition, non-zero weak solutions can originate from zero initial data, revealing non-uniqueness through convex integration techniques.

Contribution

It extends convex integration methods to hyperbolic elastodynamic systems with null conditions, showing non-uniqueness of solutions from zero initial data.

Findings

Existence of non-zero weak solutions from zero initial data

Application of convex integration to hyperbolic systems with null conditions

Revelation of complex solution structures in nonlinear elastodynamics

Abstract

We consider the Cauchy problem for the system of elastodynamic equations in two dimensions. Specifically, we focus on materials characterized by a null condition imposed on the quadratic part of the nonlinearity. We can construct non-zero weak solutions $u \in C^1([0, T] \times \mathbb{T}^2)$ that emanate from zero initial data. The proof relies on the convex integration scheme. By exploiting the characteristic double wave speeds of the equations, we construct a new class of building blocks. This work extends the application of convex integration techniques to hyperbolic systems with a null condition and reveals the rich solution structure in nonlinear elastodynamics.