568 , 2015-03-31 , Graduate School of Mathematical Sciences, The University of Tokyo
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.W e 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.