Existence of a solution to the non-steady magnetohydrodynamics-Boussinesq system with mixed boundary conditions

Abstract

In this paper we are concerned with the non-steady magnetohydrodynamics (MHD)-Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, pressure, vorticity and stress conditions together, and in the case of static pressure may include the friction type of boundary conditions more. For the magnetic field, a non-homogeneous mixed boundary conditions are given. The conditions for temperature may include Dirichlet, Neumann and Robin conditions together. For the problem involving the static pressure and stress boundary conditions for fluid it is proved that if the data of problem are small enough, material parameters are independent of temperature, then under compatibility conditions in the initial time there exists a strong solution and the solution with small norm is unique. Much effort is paid to overcome a difficulty in dealing the advection term, Lorentz force in the momentum equation of fluid and a nonlinear term in the magnetic equation under the static pressure boundary conditions on a portion of boundary where there is flux of fluid and non-homogeneous magnetic boundary conditions. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proved without smallness of the data of problem.