Recent developments in applied mathematics and mechanics: theory, experiment and practice. Devoted to the 80th anniversary of academician N.N.Yanenko

Akademgorodok, Novosibirsk, Russia, June 24 - 29, 2001

Abstracts

Russian participants

Investigation of the boundary value problem for the stationary heat and mass transfer equations

Smishliaev A.B., Tereshko D.A.

Institute of Applied Mathematics FEB RAS (Vladivostok)

The boundary-value problem for the stationary heat and mass transfer equations is considered in the bounded domain of three-dimensional space. The velocity, temperature and concentration of some substantion are unknown functions. The volume density of external forces, heat and mass sources, and the velocity, temperature and concentration on the boundary of the considered domain are given functions. As for boundary conditions for the velocity then the boundary of the considered domain is divided into three parts. At the first part we set a homogeneous boundary condition for the velocity. At the second one the homogeneous condition for the tangential component of the velocity and inhomogeneous condition for the dynamic pressure are given. At the third one the homogeneous condition for the normal component of the velocity and inhomogeneous condition for the tangential component of the vorticity are given. These boundary conditions are so-called non-standard boundary conditions for the velocity. To state the boundary conditions for the temperature and concentration we introduce two independent partititions of the boundary. At the first part of each partitition we set a homogeneous Dirichlet condition for the temperature and concentration respectively. At the second part of each partitition inhomogeneous Neumann conditions are given.

We have proved the global solvability theorem for the boundary-value problem formulated above under some assumptions on the data which have no sense of smallness. Also, we have derived the precise a priori estimates for the solution. As a consequence we have established the global existence theorem for the corresponding hydrodynamic problem with non-standard boundary conditions formulated in [1].

References
1. Conca C., Murat F. and Pironneau O. The Stokes and Navier-Stokes equations with boundary conditions involving the pressure // Japan. J. Math. 1994. V. 20. P. 279-318.

The work was supported by the Russian Found of the Fundamental Researches (codes of the projects 99-01-00214, 01-01-06067, 01-01-06070).

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Note. Abstracts are published in author's edition

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