The Mixed Boundary Value Problems for the Steady Magnetohydrodynamics-Heat System with Joule Effects

We are concerned with the steady Magnetohydrodynamics(MHD)-heat system with Joule effects under mixed boundary conditions. The boundary conditions for fluid may include the stick, pressure (or total pressure), vorticity, stress (or total stress) and friction types (Tresca slip, leak, one-sided leaks) boundary conditions together and for the electromagnetic field non-homogeneous mixed boundary conditions are given. The conditions for temperature may include non-homogeneous Dirichlet, Neumann and Robin conditions together. The viscosity, magnetic permeability, electrical conductivity, thermal conductivity and specific heat of the fluid depend on the temperature. The domain for fluid is not assumed to be simply connected. For the problem involving the static pressure and stress boundary conditions for fluid it is proved that if the parameter for buoyancy effect is small in accordance with the data of problem, a datum concerned with non-homogeneous mixed boundary conditions for magnetic field and the data of problem are small enough, then there exists a solution. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proved when the parameter for buoyancy effect is small in accordance with the data of problem, a datum concerned with non-homogeneous mixed boundary conditions for magnetic field is small, but without the auxiliary smallness of the other data of problem. In addition (Appendix), a very simple proof of the fact that vorticity quadratic form for vector fields with mixed boundary conditions is positive-definite, which has been known in a previous paper and is used in this paper, is given.