Non-uniqueness of weak solutions to the 3D Hall-MHD equations on the plane

2025/08/15 by Peng, Yi, Wang, Huaqiao, Zhang, Chenlu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.11193

Abstract

We prove the non-uniqueness of weak solutions with non-trivial magnetic fields to the 3D Hall-MHD equations on the plane in the space C⁰_t L_x² through the convex integration scheme and by constructing new errors and new intermittent flows. In particular, based on the construction of 3D intermittent flows, we obtain the 2(1)/(2)D Mikado flows through a projection onto the plane. Moreover, we prove that the constructed weak solution do not conserve the magnetic helicity and find that weak solutions of the ideal Hall-MHD equations in C^β_t,x (β>0) are the strong vanishing viscosity and resistive limit of weak solutions to the Hall-MHD equations.

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Related

• Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations
• Non-unique weak solutions in Leray-Hopf class of the 3D Hall-MHD system
• Weak solutions to the Hall-MHD equations whose singular sets in time have Hausdorff dimension strictly less than 1
• Sharp and strong non-uniqueness for the magneto-hydrodynamic equations
• On partial regularity for the 3D non-stationary Hall magnetohydrodynamics equations on the plane