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Non-uniqueness of smooth solutions of the 5D magnetohydrodynamic equations from critical data

Zipeng Chen, Song Liu, Zhaoyang Yin

Abstract

Recently, Coiculescu and Palasek \cite{Coiculescu2025} shows the non-uniqueness of solutions for the 3D incompressible Navier-Stokes equations with initial data in $BMO^{-1}$. Inspired by their breakthrough work, we develop their schemes for the incompressible magnetohydrodynamic equations and obtain a similar result in 5 dimensional case. More precisely, we construct two distinct global solutions with a initial data, which has nonvanishing velocity and magnetic fields in $BMO^{-1}(\mathbb{T}^5)$.

Non-uniqueness of smooth solutions of the 5D magnetohydrodynamic equations from critical data

Abstract

Recently, Coiculescu and Palasek \cite{Coiculescu2025} shows the non-uniqueness of solutions for the 3D incompressible Navier-Stokes equations with initial data in . Inspired by their breakthrough work, we develop their schemes for the incompressible magnetohydrodynamic equations and obtain a similar result in 5 dimensional case. More precisely, we construct two distinct global solutions with a initial data, which has nonvanishing velocity and magnetic fields in .
Paper Structure (17 sections, 21 theorems, 184 equations)

This paper contains 17 sections, 21 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Main idea of the construction
  4. Notation and parameters
  5. Organization of the paper
  6. The leading part of the solutions
  7. Construction of the Mikado potential
  8. Construction of the initial data
  9. Construction of the principal parts
  10. Construction of the perturbation
  11. Estimates on the residual
  12. Semigroup of the linearization around the principal part
  13. Fixed point argument
  14. Proof of the non-uniqueness theorem
  15. Tools from convex integration
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists divergence-free initial data $(U^0,B^0)\in BMO^{-1}$ such that the MHD equations (e:MHDe) admits two distinct global solutions

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 31 more