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Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion

Changxing Miao, Zhiwen Zhao

Abstract

In this paper, we establish the non-uniqueness of solutions to the ideal magnetohydrodynamics equations in any dimension greater than three by proving the existence of infinitely many compactly supported weak solutions. In particular, these solutions fail to conserve the total energy. Our proof relies on the differential inclusion framework tailored to the geometry of ideal MHD system, which enables the simultaneous use of Baire category method and convex integration scheme.

Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion

Abstract

In this paper, we establish the non-uniqueness of solutions to the ideal magnetohydrodynamics equations in any dimension greater than three by proving the existence of infinitely many compactly supported weak solutions. In particular, these solutions fail to conserve the total energy. Our proof relies on the differential inclusion framework tailored to the geometry of ideal MHD system, which enables the simultaneous use of Baire category method and convex integration scheme.

Paper Structure

This paper contains 6 sections, 12 theorems, 99 equations.

Table of Contents

  1. Introduction and main results
  2. Relaxation of ideal MHD and localized plane wave
  3. Convex geometry and energy growth
  4. The proofs of Theorems \ref{['Main01']} and \ref{['Main02']}
  5. Baire category method
  6. Convex integration scheme

Key Result

Theorem 1.2

For any $n\geq4$, there exist nontrivial compactly supported weak solutions $(u,b,p)\in L^{\infty}(\mathbb{R}^{n}_{x}\times\mathbb{R}_{t};\mathbb{R}^{n}\times \mathbb{R}^{n}\times\mathbb{R})$ of the ideal MHD in IMHD with $f=g=0$.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of Proposition \ref{['pro01']}
  • Lemma 3.1
  • ...and 19 more