Global regularity of two-dimensional ideal incompressible MHD
Consider the ideal incompressible magnetohydrodynamic equations on $\mathbb{R}^2$, $$\partial_t u + (u \cdot \nabla) u + \nabla p = (b \cdot \nabla) b, \qquad \partial_t b + (u \cdot \nabla) b = (b \cdot \nabla) u, \qquad \nabla \cdot u = \nabla \cdot b = 0,$$ for a velocity field $u$, a magnetic field $b$ and a pressure $p$, with both the viscosity and the magnetic resistivity exactly zero. Shao, Wang, Xu and Ge proved local existence and uniqueness of classical solutions for general initial data in $C^{1,\alpha}(\mathbb{R}^n)$, $n = 2, 3$ (2017). Bardos, Sulem and Sulem proved global existence for ideal incompressible MHD when the magnetic field is a localized perturbation of a strong constant field (1988). With partial dissipation in the velocity equation and partial magnetic diffusion acting in different directions, Cao and Wu proved global regularity of classical solutions (2011).
Problem. For every pair of divergence-free Schwartz fields $u_0, b_0 : \mathbb{R}^2 \to \mathbb{R}^2$, does the system admit a global smooth finite-energy solution with $(u, b)(0) = (u_0, b_0)$? A finite-time singularity from data in this class would answer the question negatively.
Source: Classical; see Cao and Wu (2011), introduction. Stated in this form, for Schwartz data on the plane, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, global regularity, incompressible flows, magnetohydrodynamics
References
@article{cao2011,
author = {Cao, Chongsheng and Wu, Jiahong},
title = {Global regularity for the {2D} {MHD} equations with mixed partial dissipation and magnetic diffusion},
journal = {Adv. Math.},
volume = {226},
number = {2},
year = {2011},
pages = {1803--1822},
doi = {10.1016/j.aim.2010.08.017},
eprint = {0901.2908},
archivePrefix = {arXiv},
}
@article{shao2017,
author = {Shao, Shu-Guang and Wang, Shu and Xu, Wen-Qing and Ge, Yu-Li},
title = {On the local {$C^{1,\alpha}$} solution of ideal magneto-hydrodynamical equations},
journal = {Discrete Contin. Dyn. Syst.},
volume = {37},
number = {4},
year = {2017},
pages = {2103--2113},
doi = {10.3934/dcds.2017090},
}
@article{bardos1988,
author = {Bardos, C. and Sulem, C. and Sulem, P.-L.},
title = {Longtime dynamics of a conductive fluid in the presence of a strong magnetic field},
journal = {Trans. Amer. Math. Soc.},
volume = {305},
number = {1},
year = {1988},
pages = {175--191},
doi = {10.1090/S0002-9947-1988-0920153-5},
}
@misc{aim-ideal-mhd-two-dimensions,
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 two-dimensional ideal incompressible MHD}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/079-ideal-mhd-two-dimensions.md},
note = {Entry 079 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-00599, "Global regularity of two-dimensional ideal incompressible MHD", clopen.solutions.