Estimating the convex relaxation of the ideal magnetohydrodynamics equations

2025/05/15 by Borbála Fazekas, József J. Kolumbán, Fazekas, Borbála +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2505.10230

openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/08/01

Abstract

We investigate the explicit convex relaxation of the ideal magnetohydrodynamics equations. We provide a non-trivial lower estimate on the lamination hull and an upper estimate on the Λ-convex hull, the latter providing inequalities which will be satisfied by weak limits of weak solutions of the ideal MHD equations, which serve as a model of averaged turbulent magnetohydrodynamical flows.

Citations

• The Rayleigh-Taylor instability with local energy dissipation
• Relaxation of the kinematic dynamo equations
• Magnetic helicity, weak solutions and relaxation of ideal MHD
• Localized mixing zone for Muskat bubbles and turned interfaces
• Mixing solutions for the Muskat problem with variable speed
• Dissipative Euler flows for vortex sheet initial data without distinguished sign
• A New Approach to the Rayleigh–Taylor Instability
• Degraded mixing solutions for the Muskat problem
• Mixing solutions for the Muskat problem
• Non-Uniqueness and prescribed energy for the continuity equation
• Relaxation of the incompressible porous media equation

Related

• Magnetic helicity, weak solutions and relaxation of ideal MHD
• Symmetry breaking in ideal magnetohydrodynamics: the role of the velocity
• Weak solutions of ideal MHD which do not conserve magnetic helicity
• Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations
• Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations