10007934

A model of the mathematical fluid dynamics which describes the motion of a three-dimensional viscous rotating fluid in a homogeneous gravitational field with the consideration of the salinity and heat transfer is considered in a vertical finite layer. The model is a generalization of the linearized Navier-Stokes system with the addition of the Coriolis parameter and the equations for changeable density, salinity, and heat transfer. An explicit solution is constructed and the proof of the existence and uniqueness theorems is given. The localization and the structure of the spectrum of inner waves is also investigated. The results may be used, in particular, for constructing stable numerical algorithms for solutions of the considered models of fluid dynamics of the Atmosphere and the Ocean.

[1] G. Marchuk, Mathematical Models in Environmental Problems, North Holland, Elsevier, Amsterdam, 1986.

[2] S. Sobolev, On a new problem of mathematical physics, Selected works of S.Sobolev, Springer, N.Y., 2006.

[3] V. Maslennikova, The rate of decrease for large time of the solution of a Sobolev system with viscosity, Math. USSR Sb., Vol. 21, 1973, pp. 584-606.

[4] V. Maslennikova, On the rate of decay of the vortex in a viscous fluid, Proceedings of the Steklov Institute of Mathematics, Vol. 126, 1974, pp. 47-75.

[5] V. Maslennikova, and A. Giniatoulline, Spectral properties of operators for the systems of hydrodynamics of a rotating liquid and non-uniqueness of the limit amplitude, Siberian Math. J., Vol. 29, No.5, 1988, pp. 812-824.

[6] A. Giniatoulline, An Introduction to Spectral Theory, Edwards, Philadelphia, 2005.

[7] A. Giniatoulline, On the essential spectrum of operators, 2002 Proceedings WSEAS Intl. Conf. on System Science, Appl. Mathematics and Computer Science, Rio de Janeiro: WSEAS Press, 2002, pp.1291-1295.

[8] A. Giniatoulline, On the essential spectrum of operators generated by PDE systems of stratified fluids, Intern. J. Computer Research, Vol. 12, 2003, pp. 63-72.

[9] A. Giniatoulline, and C. Rincon, On the spect-rum of normal vibrations for stratified fluids, Computational Fluid Dynamics J., Vol. 13, 2004, pp. 273-281.

[10] A. Giniatoulline, and C. Hernandez, Spectral properties of compressible stratified flows, Revista Colombiana Mat., Vol. 41, (2), 2007, pp. 333-344.

[11] A. Giniatoulline, On the strong solutions of the nonlinear viscous rotating stratified fluid, Int. J. of Mat., Computational, Phys., Electr. and Computer Engineering, Vol. 10, (10), 2016, pp. 469-475.

[12] G. Marchuk, V. Kochergin, and V. Sarkisyan, Mathematical Models of Ocean Circulation, Nauka, Novosibirsk, 1980.

[13] C.Tranter, Integral Transforms in Mathematical Physics, J. Wiley and Sons, NY, 1971.

[14] V. Maslennikova, and I. Petunin, Asymptotics for tââ in the solution of An initial-boundary value problem in the theory of internal waves, Diff. Uravneniya, Vol. 31, No. 5, 1995, pp. 823-828.

[15] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1966.

[16] F. Riesz, and B. Sz.-Nag, Functional Analysis, Fr. Ungar, N.Y., 1972.

[17] G. Grubb, and G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann., Vol. 227, 1977, pp. 247-276.

[18] S. Agmon, A. Douglis, and L. Nirenberg, Esti-mates near the boundary for solutions of elliptic differential equations. Comm. Pure and Appl. Mathematics, Vol. 17, 1964, pp. 35-92.