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This paper is concerned with the vanishing viscosity and magnetic resistivity limit for the two-dimensional steady incompressible MHD system on the half plane with no-slip boundary condition on velocity field and perfectly conducting wall condition on magnetic field. We prove the nonlinear stability of shear flows of Prandtl type with nondegenerate tangential magnetic field, but without any positivity or monotonicity assumption on the velocity field. It is in sharp contrast to the steady Navier-Stokes equations and reflects the stabilization effect of magnetic field. Unlike the unsteady MHD system, we manage the degeneracy on the boundary caused by no-slip boundary condition and obtain the estimates of solutions by introducing an intrinsic weight function and some good auxiliary functions.
As a continuation of cite{LXY}, the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on velocity field and perfect conducting boundary condition
It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schrodinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The erro
In the case of favorable pressure gradient, Oleinik proved the global existence of classical solution for the 2-D steady Prandtl equation for a class of positive data. In the case of adverse pressure gradient, an important physical phenomena is the b
Motivated by the paper by D. Gerard-Varet and E. Dormy [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equation
In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions un