Existence of Solutions to the Heat Convection Equations in a Time-dependent Domain with Mixed Boundary Conditions

J. Math. Sci. Univ. Tokyo
Vol. 22 (2015), No. 2, Page 531–568.

Kim, Tujin ; Cao, Daomin
Existence of Solutions to the Heat Convection Equations in a Time-dependent Domain with Mixed Boundary Conditions
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Abstract:
In this paper we are concerned with the initial boundary value problems of the heat convection equations in a time-dependent domain with mixed boundary conditions involving the total pressure of fluid. We obtain the existence of a weak solution to the problem. By a transformation of unknown functions and a penalty method we connect the problem to an elliptic operator equation for functions defined in the time-dependent domain. Owing to the transformation we do not need to assume that the given data are small enough. This method is also valid for the Navier-Stokes equations with the nonstandard boundary conditions.

Keywords: Heat convection equations, time-dependent domain, mixed boundary conditions, existence.

Mathematics Subject Classification (2010): 35Q35, 35A01, 76D03, 80A20.
Mathematical Reviews Number: MR3287268

Received: 2014-06-23