Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

Abstract

In this paper, we are concerned with the steady magnetohydrodynamics (MHD)‐Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, Tresca slip, leak, one‐sided leaks, pressure, vorticity, and stress conditions together, and the conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. For the electromagnetic field, mixed boundary conditions are given. Also, the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat of the fluid depend on the temperature. Special attention is paid to the meaning of non‐homogeneous mixed boundary conditions for magnetic field. For the problem involving the static pressure and stress boundary conditions for fluid, it is proven that if the data of problem are small enough, then there exists a solution and the solution with small norm is unique when the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat are independent of the temperature. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proven when a datum concerned with non‐homogeneous boundary conditions for magnetic field is small, but without smallness of other data.