Existence and Uniqueness of Time-Periodic Solutions to the Semigeostrophic Equations - Rahman, Mohammad; Wang, Kening; Zhan, Mei-Qin (Journal of Applied Nonlinear Dynamics, 2026)
In this article, we study the Semigeostrophic Equations in meteorology. These equations were introduced by Hoskins and Bretherton [1]. After suitable changes of variables, one can obtain the following coupled the Monge-Ampère/Transport problem $$\begin{aligned} \frac{\partial q}{\partial t} + J(\psi, q)=0 \\ \Delta \phi + \det(\frac{\partial^2 \phi}{\partial x_i \partial x_j})+ 1 = q onumber\\ \psi_{xx} \phi_{yy} - 2 \psi_{xy} \phi_{xy} + \psi_{yy} \phi_{xx} + \Delta \psi -\Delta \phi = 0 \end{aligned}$$ We proved the existence and uniqueness of time-periodic solution to the system.