In this talk we will recall the classical Prandtl boundary layer double-scale asymptotical expansions in the analysis of structure of fluids with the high Reynolds number in a domain with boundaries. Vanishing viscosity limit can be regarded as a direct application of Prandtl boundary layer asymptotical expansions. The Prandtl boundary layer theory includes the well-posedness of boundary layer solutions and the justification of Prandtl boundary layer ansatz etc. Motivated by one open problem in the classical book “Mathematical models in Boundary Layer Theory ”by O.A. Oleinik and V.N. Samokhin. We consider the Prandtl boundary layer theory in complex fluids, such as Magneto-hydrodynamics. The solvability of MHD boundary layer equations and the validity of Prandtl boundary layer ansatz for MHD equations in Sobolev spaces are addressed for two different cases: the tangential magnetic field dominates; the normal magnetic field dominates.
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