Local-in-time well-posedness for compressible MHD boundary layer

Abstract

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 for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for 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 that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work [26] on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.2010 Mathematics Subject Classification. 35A07, 35M33, 35Q35, 76N20, 76W05.

Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces

“…Motivated by the fifteenth open problem in Oleinik-Samokhin's classical book [33] (page 500-503), "15. For the equations of the magnetohydrodynamic boundary layer, all problems of the above type are still open," efforts have been made to study the well-posedness of solutions to the MHD boundary layer equations and to justify the MHD boundary layer ansatz in [12,23,24,39,40]; see also [9]. Precisely, when the hydrodynamic and magnetic Reynolds numbers have the same order, the well-posedness of solutions to the MHD boundary layer equations and the validity of the Prandtl ansatz were established without any monotone condition imposed on the velocity in [23,24].…”

Global strong solution to 3D full compressible magnetohydrodynamic flows with vacuum at infinity

“…The one-dimension problem has been studied in many papers, for example, [3,4,8,9,18,32] and the references cited therein. For the multi-dimensional case, see [6,10,[12][13][14][15][16][17][22][23][24][25]27,29] and so on. To be more specific, for the 2D case, Cao and Wu [2] obtained the global existence, conditional regularity and uniqueness of a weak solution with only magnetic diffusion.…”

Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow

“…Boundary layer problem on some more complex fluids such as magnetohydrodynamic was also made great progresses, cf. [12,18,19,20,24,25,26,32,33,38,39,41]. The main object in this paper is to establish the local well-posedness of the two-phase boundary layer equations, which are derived from 2D two-phase flow with non-slip boundary condition on the velocity.…”