In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain {(X, Y ) ∈ [0, L] × R + } with a moving flat boundary {Y = 0}. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the same order in term of the reciprocal of a small parameter ǫ, the tangential component of the magnetic field does not degenerate near the boundary and the ratio of the strength of tangential component of magnetic field and tangential component of velocity is suitably small. And the error terms are estimated in L ∞ sense.
In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow (1 − y 2 , 0) in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate e −c √ νt . As an application, we prove that if the initial perturbation of vortiticy satisfiesfor some small constant c 0 > 0 independent of the viscosity ν, then the solution dose not transition away from the Poiseuille flow for any time.Here ω = ∂ y u 1 − ∂ x u 2 is the vorticity.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.