Global regularity of viscous planar MHD without magnetic diffusion
Consider the incompressible magnetohydrodynamics equations on $\mathbb{R}^2$ with unit viscosity and no magnetic diffusion or damping, $$\partial_t u + u \cdot \nabla u - \Delta u + \nabla P = b \cdot \nabla b, \qquad \partial_t b + u \cdot \nabla b = b \cdot \nabla u, \qquad \nabla \cdot u = \nabla \cdot b = 0,$$ for the velocity $u$, the magnetic field $b$ and the total pressure $P$. Fefferman, McCormick, Robinson and Rodrigo proved that for divergence-free data in $H^s(\mathbb{R}^2)$, $s > 1$, there is a unique solution in $C([0, T_*]; H^s)$ for some $T_* > 0$. Lin, Xu and Zhang, and Ren, Wu, Xiang and Zhang, proved global existence of smooth solutions for data close to an equilibrium with a nonzero constant magnetic field.
Problem. For every pair of divergence-free $u_0, b_0 \in \bigcap_{k \ge 0} H^k(\mathbb{R}^2; \mathbb{R}^2)$, does the system with $(u, b)(0) = (u_0, b_0)$ have a unique solution $(u, b) \in C([0, \infty); H^k \times H^k)$ for every integer $k \ge 0$? No smallness, symmetry or imposed constant background magnetic field is allowed.
Source: A long-standing question; see Lin, Xu and Zhang (2015), §1, and Fefferman, McCormick, Robinson and Rodrigo (2014), §1. Stated in this form by AIM — Open Applied Problems, after Suo and Jiu (2022), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, global regularity, magnetohydrodynamics
References
@article{lin2015,
author = {Lin, Fanghua and Xu, Li and Zhang, Ping},
title = {Global small solutions of 2-{D} incompressible {MHD} system},
journal = {J. Differential Equations},
volume = {259},
number = {10},
year = {2015},
pages = {5440--5485},
doi = {10.1016/j.jde.2015.06.034},
eprint = {1302.5877},
archivePrefix = {arXiv},
}
@article{fefferman2014,
author = {Fefferman, Charles L. and McCormick, David S. and Robinson, James C. and Rodrigo, Jose L.},
title = {Higher order commutator estimates and local existence for the non-resistive {MHD} equations and related models},
journal = {J. Funct. Anal.},
volume = {267},
number = {4},
year = {2014},
pages = {1035--1056},
doi = {10.1016/j.jfa.2014.03.021},
eprint = {1401.5018},
archivePrefix = {arXiv},
}
@article{suo2022,
author = {Suo, Xiaoxiao and Jiu, Quansen},
title = {Global well-posedness of {2D} incompressible magnetohydrodynamic equations with horizontal dissipation},
journal = {Discrete Contin. Dyn. Syst.},
volume = {42},
number = {9},
year = {2022},
pages = {4523--4553},
doi = {10.3934/dcds.2022063},
}
@article{ren2014,
author = {Ren, Xiaoxia and Wu, Jiahong and Xiang, Zhaoyin and Zhang, Zhifei},
title = {Global existence and decay of smooth solution for the 2-{D} {MHD} equations without magnetic diffusion},
journal = {J. Funct. Anal.},
volume = {267},
number = {2},
year = {2014},
pages = {503--541},
doi = {10.1016/j.jfa.2014.04.020},
}
@misc{aim-nonresistive-planar-mhd,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global regularity of viscous planar MHD without magnetic diffusion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/370-nonresistive-planar-mhd.md},
note = {Entry 370 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00626, "Global regularity of viscous planar MHD without magnetic diffusion", clopen.solutions.