No abstract

This paper is concerned with the validity of Prandtl boundary layer expansions for 2D steady viscous incompressible magnetohydrodynamic (MHD) flows over a rotating disk {( θ, r) ∈ [0, θ0] × [ R0, ∞)} with a moving curved boundary { r = R0}. We establish the validity of boundary layer expansions and convergence rates in the Sobolev sense. Then, we extend the results by Iyer [Arch. Ration. Mech. Anal. 224(2), 421–469 (2017)] for Navier–Stokes equations to the MHD flows.

In this paper, we justify the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear parallel pipe flow of nonhomogeneous incompressible Navier-Stokes equations. The convergence for velocity is shown under various Sobolev norms. In addition, the higher-order asymptotic expansions are also considered. And the mathematical validity of the Prandtl boundary layer theory for nonlinear parallel pipe flow is generalized to the nonhomogeneous case.