Non-uniqueness for the nonlinear dynamical Lam\'e system

TL;DR

This paper demonstrates the non-uniqueness of solutions for the nonlinear dynamical Lamé system in 2D and 3D, using convex integration to construct infinitely many solutions from the same initial data.

Contribution

It introduces a novel convex integration approach with new building blocks tailored for the Lamé system with double wave speeds, showing non-uniqueness of solutions.

Findings

Infinitely many solutions can emanate from the same initial data.

Solutions are in the class C^{1,α} with α<1/60.

The method applies to both 2D and 3D cases.

Abstract

We consider the Cauchy problem for the nonlinear dynamical Lam\'e system with double wave speeds in a $d$-dimensional $(d=2,3)$ periodic domain. Moreover, the equations can be transformed into a linearly degenerate hyperbolic system. We could construct infinitely many continuous solutions in $C^{1,\alpha}$ emanating from the same small initial data for $\alpha<\frac{1}{60}$. The proof relies on the convex integration scheme. We construct a new class of building blocks with compression structure by using the double wave speeds characteristic of the equations.