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Small-time existence of a strong solution of primitive equations for the ocean. (English) Zbl 1270.35326
The idea of weather forecast was given by V. Bjerknes in [Meteorol. Zs. 21, 1–7 (1904; JFM 35.0968.03)] and then the numerical weather forecast was executed by L. F. Richarson in [Weather prediction by numerical process. London: Cambridge Univ. Press (1922; JFM 48.0629.07)]. He gave a system of equations describing the motion of atmosphere very similar to the Navier-Stokes equations.
In this paper such primitive equations as the model equations describing the motion of atmosphere, ocean and coupled atmosphere and ocean are under consideration. The authors discuss in details the free boundary problem of the primitive equations for the ocean in three-dimensional strip with surface tension.They prove temporally local existence of the unique strong solution to the transformed problem in Sobolev-Slobodetskii spaces. For this purpose the authors use the so-called $$p$$-coordinates and a coordinate transformation similar to that in [J. T. Beale, Arch. Ration. Mech. Anal. 84, 307–352 (1984; Zbl 0545.76029)] – in such way they fix the time-dependent domain.

##### MSC:
 35M10 PDEs of mixed type 35Q35 PDEs in connection with fluid mechanics 35R35 Free boundary problems for PDEs 76D99 Incompressible viscous fluids 86A05 Hydrology, hydrography, oceanography 86A10 Meteorology and atmospheric physics
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