Existence and Uniqueness of Time-Periodic Solutions to the Semigeostrophic Equations
DOI:
https://doi.org/10.5890/JAND.2025.06.003Abstract
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 $$\begin{aligned} \frac{\partial q}{\partial t} + J(\psi, q)=0 \\ \Delta \phi + \det(\frac{\partial^2 \phi}{\partial x_i \partial x_j})+ 1 = q\nonumber\\ \psi_{xx} \phi_{yy} - 2 \psi_{xy} \phi_{xy} + \psi_{yy} \phi_{xx} + \Delta \psi -\Delta \phi = 0 \end{aligned}$$ We proved the existence and uniqueness of time-periodic solution to the system.References
[1] Gilberg, D. and Trudinger, N.S. (1982), Elliptic Partial Differential Equations of Second Order, Springer-Verlag.
[2] Galdi, G.P. (2013), Existence and uniquness of time-periodic solutionsto the Navier-Stokes euqations in the whole plane, Disrete and Continuous Dynamical Systems Series S, 6(5), 1237-1257.
[3] Hoskins, B.J. and Bretherton, F.P. (1972), Atomspheric frontogenesis models: mathematical formulation and solution, Journal of the Atmospheric Sciences, 29(1), 11-37.
[4] Amann, H. (1990), Ordinary Differentiations: An Introduction to Nonlinear Analysis, Walter de Gruyter, 1450-1461.
[5] Zhan, M. (2000), Existence and uniqueness of classical solutions to semigeostrophic equations, Applicable Analysis, 75(1-2), 175-181.
[6] Galdi, G.P. and Kyed, M. (2018), Time-periodic solutions to the Navier-Stokes euqations in the whole plane, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 6(5), 509-578.
[7] Zhan, M. (2000), Existence of periodic solutions for Ginzburg-Landau equations of superconductivity, Journal of Mathematical Analysis and Applications, 249(2), 614-625.
[8] Zhan, M. (2008), Multiplicity and stability of time-periodic solutions of Ginzburg-Landau equations of superconductivity, Journal of Mathematical Analysis and Applications, 340(1), 126–134.
[9] Wang, K. and Zhan, M. (2024), Attractivity of time-periodic solutions of Ginzburg-Landau equations of superconductivity and numerical simulations, Journal of Vibration Testing and System Dynamics, 8, 317-328.
[10] Galdi, G.P. and Sohr, H. (2004), Existence and uniquness of time-periodic solutions physically reasonable Navier-Stokes flow past a body, Archive for Rational Mechanics and Analysis, 172, 363-406.
[11] Alaa, N. and Iguernane, M. (2002), Weak Periodic Solution of Some Quasilinear Parabolic Equations With Data Measure, Journal of Inequalities in Pure and Applied Mathematics, 3, Article 46.
[12] Zhan, M. Positive Time and Almost Time Periodic Solutions for the Quasigeostrophic Motions, Journal of Discontinuity, Nonlinearity, and Complexity, Accepted.
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