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The null condition in elastodynamics leads to non-uniqueness

Shunkai Mao, Peng Qu

TL;DR

This work establishes nonuniqueness for the 2D elastodynamic Cauchy problem by applying convex integration under a weak null condition on the stored-energy. The authors develop a tailored local inverse of divergence in Lagrangian coordinates to absorb Reynolds stress, leveraging the system's double wave speeds through novel building blocks and a three-direction perturbation scheme. They construct a convergent sequence of approximate solutions with vanishing Reynolds stress, yielding a $C^{1}$ weak solution emanating from zero data, thereby revealing rich nonlinear elastodynamics behavior. This extends convex integration methods to hyperbolic elastodynamic systems with null-type structures and highlights the role of weak null conditions in enabling nonuniqueness and complex solution landscapes.

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.

The null condition in elastodynamics leads to non-uniqueness

TL;DR

This work establishes nonuniqueness for the 2D elastodynamic Cauchy problem by applying convex integration under a weak null condition on the stored-energy. The authors develop a tailored local inverse of divergence in Lagrangian coordinates to absorb Reynolds stress, leveraging the system's double wave speeds through novel building blocks and a three-direction perturbation scheme. They construct a convergent sequence of approximate solutions with vanishing Reynolds stress, yielding a weak solution emanating from zero data, thereby revealing rich nonlinear elastodynamics behavior. This extends convex integration methods to hyperbolic elastodynamic systems with null-type structures and highlights the role of weak null conditions in enabling nonuniqueness and complex solution landscapes.

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 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.

Paper Structure

This paper contains 31 sections, 18 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. Basic concepts and symbols in elasticity mechanics
  3. Structure assumptions and the main result
  4. Difficulties and strategies
  5. Organization of the paper
  6. Outline of the induction scheme
  7. Mollification
  8. Cutoffs
  9. Defining a local inverse of divergence by solving the asymmetric divergence equation
  10. Solving the symmetric divergence equation
  11. Derivation of the solution operator
  12. Formula for $R_{lj}[U]$ and basic properties
  13. Estimates for the solution operator and proof of Lemma \ref{['lm: sym div eq']}
  14. Solving the asymmetric divergence equation
  15. Some notations
  16. ...and 16 more sections

Key Result

Theorem 1.7

Assume that the stored-energy function $\sigma$ in stored-energy function satisfies Weak null condition, we can construct nontrivial weak solutions $u \in C^{1}([0, T] \times \mathbb{T}^2)$ to system 1 that emanates from zero initial data.

Theorems & Definitions (51)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3: Weak Solution
  • Definition 1.4: Weak null condition
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2: Geometric Lemma
  • proof
  • ...and 41 more