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We consider the validity of Prandtl boundary layer expansion of solutions tothe initial boundary value problem for inhomogeneous incompressiblemagnetohydrodynamics (MHD) equations in the half plane when both viscosity andresistivity coefficients tend to zero, where the no-slip boundary condition isimposed on velocity while the perfectly conducting condition is given onmagnetic field. Since there exist strong boundary layers, the essentialdifficulty in establishing the uniform $L^\infty$ estimates of the errorfunctions comes from the unboundedness of curl of the strong boundary layers.Under the assumptions that the viscosity and resistivity coefficients take thesame order of a small parameter and the initial tangential magnetic field has apositive lower bound near the boundary, we prove the validity of Prandtl ansatzin $L^\infty$ sense in Sobolev framework. Compared with the homogeneousincompressible case considered in \cite{LXY192}, some suitable functionalsshould be designed and the elaborated co-normal energy estimates will beinvolved in analysis due to the variation of density and the interactionbetween the density and velocity.

Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework