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Existence theorems for a three-dimensional ocean dynamics model and a data assimilation problem. (English. Russian original) Zbl 1327.35309
Dokl. Math. 75, No. 1, 28-30 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 412, No. 2, 151-153 (2007).
From the text: The solvability of the three-dimensional equations governing large-scale ocean dynamics was analyzed in [G. I. Marchuk and B. A. Kagan, Dynamics of ocean. Dordrecht: Kluwer (1989); J.-L. Lions et al., Nonlinearity 5, No. 5, 1007–1053 (1992; Zbl 0766.35039); R. Temam and M. Ziane, in: Handbook of mathematical fluid dynamics. Vol. III. Amsterdam: Elsevier/North Holland. 535–567 (2004; Zbl 1222.35145)] under the assumption that the vertical velocity of the water at the upper boundary of the domain is zero. In this paper, we establish the existence of solutions for a model in which this assumption is not used and the vertical velocity at the ocean’s free surface is assumed to be a variable defined in terms of the sea level height. In the second part of this work, we prove the solvability of a variational data assimilation problem [V. I. Agoshkov, Optimal control methods and the method of adjoint equations in problems of mathematical physics (Russian). Moskva: Rossiń≠skaya Akademiya Nauk (2003; Zbl 1236.49002); V. I. Agoshkov and V. M. Ipatova, Dokl. Math. 57, No. 3, 404–406 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 360, No. 4, 439–441 (1998; Zbl 0964.86001); N. N. Bogolyubov, Collection of scientific works, Vol. 1–3 (Russian). Moskva: Nauka (2005; Zbl 1198.01033; Zbl 1198.01032; Zbl 1198.01034); V. I. Agoshkov et al., Russ. J. Numer. Anal. Math. Model. 20, No. 1, 19–43 (2005; Zbl 1067.65099)].
MSC:
35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
93B30 System identification
93C20 Control/observation systems governed by partial differential equations
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References:
[1] V. I. Agoshkov, Optimal Control Methods and Adjoint Equations in Mathematical Physics Problems (Inst. Vychisl. Mat. Ross. Akad. Nauk, Moscow, 2003) [in Russian].
[2] V. I. Agoshkov and V. M. Ipatova, Dokl. Math. 57, 404–406 (1998) [Dokl. Akad. Nauk 360, 439–441 (1998)].
[3] N. N. Bogolyubov, Collected Scientific Works (Nauka, Moscow, 2005) [in Russian].
[4] G. I. Marchuk and B. A. Kagan, Dynamics of Ocean (Kluwer Academic, Dordrecht, 1989; Gidrometeoizdat, Leningrad, 1989).
[5] V. I. Agoshkov, F. P. Minuk, A. S. Rusakov, and V. B. Zalesny, Russ. J. Numer. Anal. Math. Model., No. 1, 19–43 (2005).
[6] J.-L. Lions, R. Temam, and S. Wang, Nonlinearity 5, 1007–1053 (1992). · Zbl 0766.35039 · doi:10.1088/0951-7715/5/5/002
[7] R. Temam and M. Ziane, Some Mathematical Problems in Geophysical Fluid Dynamics: Handbook of Mathematical Fluid Dynamics (Elsevier, Amsterdam, 2004), Vol. 3. · Zbl 1222.35145
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