Vanishing dissipation limit of solutions to initial boundary value problem for three dimensional incompressible magneto-hydrodynamic equations with transverse magnetic field

Abstract

In this paper, we study the vanishing dissipation limit of solutions to the initial boundary value problem of three-dimensional incompressible viscous magnetohydrodynamic equations in the upper half space where the magnetic field is assumed to be transverse to the boundary. When the tangential component of the magnetic field is imposed the zero Neumann boundary condition, we can establish the uniform energy estimates of the solutions to the initial boundary value problem in the conormal Sobolev spaces even if the velocity field satisfies the no-slip boundary condition. Then the vanishing dissipation limit of solutions to the initial boundary value problem can be achieved in L∞ sense based on some compactness arguments. Compared with the previous results, our results show that the strong boundary layer can still be prevented by the transverse magnetic field even with the magnetic diffusion.