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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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##### References:
 [1] Agoshkov, V.I.; Ipatova, V.M., Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of Ocean circulation, Comput. math. math. phys., 37, 3, 348-358, (1997) · Zbl 0954.49029 [2] Bernier, C., Existence of attractor for the quasi-geostrophic approximation of the navier – stokes equations and estimate of its dimension, Adv. math. sci. appl., 4, 2, 465-489, (1994) · Zbl 0815.76086 [3] Bernier-Kazantsev, C.; Chueshov, I.D., The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer Ocean model, Nonlinear anal., 42, 1499-1512, (2000) · Zbl 0996.86002 [4] Colin, T., The Cauchy problem and the continuous limit for the multilayer model in geophysical fluid dynamics, SIAM J. math. anal., 28, 3, 516-529, (1997) · Zbl 0879.76115 [5] Haltiner, G.J.; Williams, R.T., Numerical prediction and dynamic meteorology, (1980), John Wiley and Sons New York [6] Hu, C., Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear anal., 61, 3, 425-460, (2005) · Zbl 1081.35080 [7] Hu, C.; Temam, R.; Ziane, M., Regularity results for linear elliptic problems related to the primitive equations, Chinese ann. math., 23B, 2, 1-16, (2002) [8] Hu, C.; Temam, R.; Ziane, M., The primitive equations of the large scale Ocean under the small depth hypothesis, Discrete contin. dyn. syst., 9, 1, 97-131, (2003) · Zbl 1048.35082 [9] Lions, J.L.; Temam, R.; Wang, S., On the equations of large-scale Ocean, Nonlinearity, 5, 1007-1053, (1992) · Zbl 0766.35039 [10] Lions, J.L.; Temam, R.; Wang, S., Models of the coupled atmosphere and Ocean (CAOI), Comput. mech. adv., 1, 3-54, (1993) · Zbl 0805.76011 [11] Lions, J.L.; Temam, R.; Wang, S., Numerical analysis of the coupled atmosphere and Ocean models (CAOII), Comput. mech. adv., 1, 55-120, (1993) · Zbl 0805.76052 [12] Lions, J.L.; Temam, R.; Wang, S., Mathematical study of the coupled models of atmosphere and Ocean (CAOIII), Math. pures appl., 73, 105-163, (1995) [13] Lions, J.L.; Temam, R.; Wang, S., On mathematical problems for the primitive equations of the Ocean: the mesoscale midlatitude case. lakshmikantham’s legacy: A tribute on his 75th birthday, Nonlinear anal. TMA, 40, 1-8, 439-482, (2000) · Zbl 0978.76102 [14] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005 [15] Peixoto, J.P.; Oort, A.H., Physics of climate, (1992), American Institute of Physics New York [16] Temam, R., () [17] Washington, W.M.; Parkinson, C.L., An introduction to three-dimensional climate modeling, (1986), Oxford University Press Oxford · Zbl 0655.76003
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