Global smooth evolution of small semigeostrophic perturbations
On $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$ consider the semigeostrophic system in dual variables, $$\partial_t \rho + \nabla \cdot (\rho U) = 0, \qquad U = J(x - \nabla P^*), \qquad \det D^2 P^* = \rho,$$ where $J(a, b) = (-b, a)$, $P^*(t, \cdot)$ is convex and $P^*(t, x) - |x|^2/2$ is periodic with mean zero. Loeper proved that if $\rho_0$ is Hölder continuous and bounded above and below by positive constants, there is a unique Hölder continuous solution on a time interval depending only on these bounds and the Hölder norm of $\rho_0$; smoother data give correspondingly smoother solutions. Ambrosio, Colombo, De Philippis and Figalli proved global-in-time existence of distributional solutions of the semigeostrophic equations in physical space on the two-dimensional torus, under mild assumptions on the initial data. The dual system is analogous to the two-dimensional incompressible Euler equations, whose smooth solutions exist for all time.
Problem. Are there an integer $k \ge 2$ and an $\varepsilon > 0$ such that every $\rho_0 \in C^\infty(\mathbb{T}^2)$ with $\rho_0 > 0$, $\int_{\mathbb{T}^2} \rho_0 = 1$ and $\|\rho_0 - 1\|_{C^k} < \varepsilon$ produces a solution that is smooth for all $t \ge 0$? Smoothness, with $D^2 P^*$ positive definite, is required on every finite time interval; no bound uniform in time is required.
Source: De Philippis and Figalli (2014), §5.2, open problem (3), who ask for conditions on the initial data ensuring global smooth solutions in the two-dimensional periodic case, and Figalli (2018), §6, problem 2, for data sufficiently close to 1 in some strong norm; stated with closeness in a C^k norm 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 smooth solutions, Monge–Ampère equations, semigeostrophic equations
References
@article{dephilippis2014,
author = {De Philippis, Guido and Figalli, Alessio},
title = {The {M}onge--{A}mp{\`e}re equation and its link to optimal transportation},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {51},
number = {4},
year = {2014},
pages = {527--580},
doi = {10.1090/S0273-0979-2014-01459-4},
eprint = {1310.6167},
archivePrefix = {arXiv},
}
@incollection{figalli2018,
author = {Figalli, Alessio},
title = {Global existence for the semigeostrophic equations via {S}obolev estimates for {M}onge--{A}mp{\`e}re},
booktitle = {Partial Differential Equations and Geometric Measure Theory},
editor = {Farina, Alberto and Valdinoci, Enrico},
series = {Lecture Notes in Math.},
volume = {2211},
publisher = {Springer},
address = {Cham},
year = {2018},
pages = {1--42},
doi = {10.1007/978-3-319-74042-3_1},
}
@article{loeper2006,
author = {Loeper, Gr{\'e}goire},
title = {A fully nonlinear version of the incompressible {E}uler equations: the semigeostrophic system},
journal = {SIAM J. Math. Anal.},
volume = {38},
number = {3},
year = {2006},
pages = {795--823},
doi = {10.1137/050629070},
eprint = {math/0504138},
archivePrefix = {arXiv},
}
@article{ambrosio2012,
author = {Ambrosio, Luigi and Colombo, Maria and De Philippis, Guido and Figalli, Alessio},
title = {Existence of {E}ulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case},
journal = {Comm. Partial Differential Equations},
volume = {37},
number = {12},
year = {2012},
pages = {2209--2227},
doi = {10.1080/03605302.2012.669443},
eprint = {1111.7202},
archivePrefix = {arXiv},
}
@misc{aim-semigeostrophic-small-data-global-smoothness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global smooth evolution of small semigeostrophic perturbations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/388-semigeostrophic-small-data-global-smoothness.md},
note = {Entry 388 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-00897, "Global smooth evolution of small semigeostrophic perturbations", clopen.solutions.