In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work , we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.
In this paper, we prove the global existence of solutions with analytic regularity to the 2D magnetohydrodynamic (MHD) boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multiscale expansion in [D. Gerard-Varet and M. Prestipino, <i>Z. Angew. Math. Phys.</i>, 68 (2017), 76]. The analysis shows that the combined effect of the magnetic diffusivity and transverse magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global-in-time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.
The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity . Under some conditions on the initial and boundary data, we show that the thickness is of the order | In |. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.