Differential and Integral Equations

Remarks on the planetary geostrophic model of gyre scale ocean circulation

R. Samelson, R. Temam, and S. Wang

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Abstract

We study in this article the mathematical formulation of the planetary geostrophic (PG) equations of large-scale ocean circulation, in the case where small-scale processes are parameterized by the traditional Laplacian eddy diffusion and eddy viscosity. We prove the existence and uniqueness of global in time strong solutions of these equations with either $L^\infty$ or $H^2$ initial data. Due essentially to the high nonlinearity (comparable to a squared gradient) of the equations, two problems remain open. First, the existence of more regular solutions with $L^\infty \cap H^1$ initial data is still unknown, although more regular solutions are obtained with $H^2$ initial data. Second, the existence of global attractor and its dimension estimates are open, and related to that are the time uniform boundedness of the norm in $H^2$ and higher order Sobolev spaces of the solutions.

Article information

Source
Differential Integral Equations Volume 13, Number 1-3 (2000), 1-14.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356124287

Mathematical Reviews number (MathSciNet)
MR1811946

Zentralblatt MATH identifier
0979.35118

Subjects
Primary: 35Q80: PDEs in connection with classical thermodynamics and heat transfer
Secondary: 35A05 35Q35: PDEs in connection with fluid mechanics 86A10: Meteorology and atmospheric physics [See also 76Bxx, 76E20, 76N15, 76Q05, 76Rxx, 76U05]

Citation

Samelson, R.; Temam, R.; Wang, S. Remarks on the planetary geostrophic model of gyre scale ocean circulation. Differential Integral Equations 13 (2000), no. 1-3, 1--14. https://projecteuclid.org/euclid.die/1356124287.


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