Non-uniqueness for the nonlinear dynamical Lamé system

TL;DR

This paper proves non-uniqueness of weak solutions for the nonlinear dynamical Lamé system on $[0,T]\times\mathbb{T}^d$ with double wave speeds in dimensions $d=2,3$ by employing a convex integration scheme. It develops a new class of building blocks that couple longitudinal and transverse waves, exploiting the speed gap $(\lambda+\mu)>0$ to control the Reynolds stress. The main results show the existence of infinitely many distinct weak solutions in $C^{1,\alpha}([0,T]\times\mathbb{T}^3)$ for any $0<\alpha<\frac{1}{60}$ (and $C^{1,\alpha}([0,T]\times\mathbb{T}^2)$ for $0<\alpha<\frac{1}{30}$ in 2D), all emanating from the same small initial data. The approach extends convex integration to elastodynamic-type systems with linear degeneracy and a null-form nonlinearity, highlighting the role of double-wave-speed structure in enabling non-uniqueness.

Abstract

We consider the Cauchy problem for the nonlinear dynamical Lamé 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 $C^{1,α}$ emanating from the same small initial data for $α<\frac{1}{60}$. 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.

Theorems & Definitions (24)

• Definition 1.1: Weak Solution
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2: Geometric Lemma
• Proposition 2.3: Inductive proposition
• Proposition 2.4: Bifurcating inductive proposition
• Remark 3.1
• Remark 3.2