Smooth unforced inviscid Boussinesq evolution on the whole plane
Consider the two-dimensional Boussinesq system without viscosity or thermal diffusion on the whole plane, $$\partial_t u + (u \cdot \nabla) u + \nabla p = \theta e_2, \qquad \partial_t \theta + u \cdot \nabla \theta = 0, \qquad \nabla \cdot u = 0,$$ where $e_2 = (0,1)$, with no external forcing beyond the displayed buoyancy coupling and no solid boundary. Chae proved global regularity when exactly one of the viscosity and the thermal diffusivity is zero (2006). Elgindi and Jeong proved finite-time singularity formation for finite-energy data with Lipschitz velocity and density, smooth away from the vertex, on wedges $\{x_1 \ge \gamma |x_2|\}$ (2020). Chen and Hou proved finite-time blowup with $C^{1,\alpha}$ velocity in the presence of a boundary (2021), and finite-time, nearly self-similar blowup for smooth initial data of finite energy in the presence of a boundary, with computer-assisted estimates (2022; 2025).
Problem. For arbitrary real Schwartz data $u_0 : \mathbb{R}^2 \to \mathbb{R}^2$ and $\theta_0 : \mathbb{R}^2 \to \mathbb{R}$ with $\nabla \cdot u_0 = 0$, does the system have a global smooth finite-energy solution with $(u, \theta)(0) = (u_0, \theta_0)$? A finite-time singularity from such data would answer the question negatively.
Source: Classical; Chae (2006), introduction, calls the regularity/singularity question for the two-dimensional Boussinesq system with zero viscosity and zero diffusivity an outstanding open problem. Stated in this form, on the whole plane with Schwartz data and no forcing, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Boussinesq equations, finite-time blowup, global regularity
References
@article{chae2006,
author = {Chae, Dongho},
title = {Global regularity for the {2D} {B}oussinesq equations with partial viscosity terms},
journal = {Adv. Math.},
volume = {203},
number = {2},
year = {2006},
pages = {497--513},
doi = {10.1016/j.aim.2005.05.001},
}
@article{elgindi2020,
author = {Elgindi, Tarek M. and Jeong, In-Jee},
title = {Finite-time singularity formation for strong solutions to the {B}oussinesq system},
journal = {Ann. PDE},
volume = {6},
number = {1},
year = {2020},
pages = {5},
doi = {10.1007/s40818-020-00080-0},
eprint = {1708.02724},
archivePrefix = {arXiv},
}
@article{chen2021,
author = {Chen, Jiajie and Hou, Thomas Y.},
title = {Finite time blowup of {2D} {B}oussinesq and {3D} {E}uler equations with {$C^{1,\alpha}$} velocity and boundary},
journal = {Comm. Math. Phys.},
volume = {383},
number = {3},
year = {2021},
pages = {1559--1667},
doi = {10.1007/s00220-021-04067-1},
eprint = {1910.00173},
archivePrefix = {arXiv},
}
@misc{chen2022,
author = {Chen, Jiajie and Hou, Thomas Y.},
title = {Stable nearly self-similar blowup of the {2D} {B}oussinesq and {3D} {E}uler equations with smooth data {I}: Analysis},
year = {2022},
eprint = {2210.07191},
archivePrefix = {arXiv},
note = {Revised 2026 (v4)},
}
@article{chen2025,
author = {Chen, Jiajie and Hou, Thomas Y.},
title = {Stable nearly self-similar blowup of the {2D} {B}oussinesq and {3D} {E}uler equations with smooth data {II}: Rigorous numerics},
journal = {Multiscale Model. Simul.},
volume = {23},
number = {1},
year = {2025},
pages = {25--130},
doi = {10.1137/23M1580395},
eprint = {2305.05660},
archivePrefix = {arXiv},
}
@misc{aim-inviscid-boussinesq-plane,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Smooth unforced inviscid Boussinesq evolution on the whole plane}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/082-inviscid-boussinesq-plane.md},
note = {Entry 082 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-00603, "Smooth unforced inviscid Boussinesq evolution on the whole plane", clopen.solutions.