Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion

2025/09/08 by Changxing Miao, Miao, Changxing, Zhao, Zhiwen
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.06866

openalex publication_date 2025/09/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

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.

Citations

• Magnetic helicity, weak solutions and relaxation of ideal MHD
• Weak solutions of ideal MHD which do not conserve magnetic helicity
• Convex integration and phenomenologies in turbulence
• A basis of Casimirs in 3D magnetohydrodynamics
• Nonuniqueness of weak solutions to the Navier-Stokes equation
• Onsager's conjecture for admissible weak solutions
• High dimensionality and h-principle in PDE
• A Proof of Onsager's Conjecture
• Dissipative Euler Flows and Onsager's Conjecture
• Convex integration for Lipschitz mappings and counterexamples to regularity
• Statistical hydrodynamics

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