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On strong solutions of the multi-layer quasi-geostrophic equations of the ocean. (English) Zbl 1157.35461
Summary: We study the multi-layer quasi-geostrophic equations of the ocean. The existence of strong solutions is proved. We also prove the existence of a maximal attractor in \(L^{2}(\varOmega )\) and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our estimates rely on a new formulation that we introduce for the multi-layer quasi-geostrophic equation of the ocean, which replaces the nonhomogeneous boundary conditions (and the nonlocal constraint) on the stream-function by a simple homogeneous Dirichlet boundary condition. This work improves the results given in [C. Bernier, Adv. Math. Sci. Appl. 4, No. 2, 465–489 (1994; Zbl 0815.76086)].

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
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