Justification of Prandtl Ansatz for MHD Boundary Layer

Abstract

As a continuation of [43], 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 on magnetic field. Under the assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a L 8 estimate on the error by multi-scale analysis.(1. 4)2000 Mathematics Subject Classification. 76N20, 35A07, 35G31, 35M33.

Lifespan of Solutions to MHD Boundary Layer Equations with Analytic Perturbation of General Shear Flow

“…Since the sovability of the system (1.4) requires only the normal components of the velocity and magnetic fields (v 0 e , g 0 e ) on the boundary (v 0 e , g 0 e )| Y =0 = 0, (1.5) in the limit from (1.1) to (1.4), a Prandtl-type boundary layer can be derived to resolve the mismatch of the tangential components between the viscous flow (u ǫ , h ǫ ) and invicid flow (u 0 , h 0 ) on the boundary {Y = 0}. And this system governing the fluid behavior in the leading order of approximation near the boundary is derived in [7,12,13]: (1.6) in H = {(x, y) ∈ R 2 |y ≥ 0} with the fast variable y = Y / √ ǫ. Here, the trace of the horizontal ideal MHD flow (1.4) on the boundary {Y = 0} is assumed to be a constant vector so that the pressure term ∂ x p 0 e (t, x, 0) vanishes by the Bernoulli's law.…”

“…On the other hand, to study the high Reynolds number limits for the MHD equations (1.1) with no-slip boundary condition on the velocity (1.2) and perfect conducting boundary condition (1.3) on the magnetic field, one can apply the Prandtl ansatz to derive the boundary layer system (1.6) as the leading order description on the flow near the boundary. For this, readers can refer to [7,12,13,14,21] about the formal derivation of (1.6), the well-posedness theory of the system and the justification of the Prandtl ansatz locally in time.…”

Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations

“…Recently, in homogeneous case, the convergence results without the compatibility conditions for some symmetric flows were established by Gie et al [7]. In addition, the validity of the boundary layer theory for incompressible MHD was obtained by Liu, Xie and Yang [17].…”

Validity of boundary layer theory for the 3D plane‐parallel nonhomogeneous electrically conducting flows

“…With this intention in mind, there are some literatures on the boundary layer theory of the MHD equations. Under the assumption of nondegenerate tangential magnetic field, Liu, Xie and Yang constructed the well‐posedness theory 25 of boundary layer equations and the convergence theory 26 for boundary layer expansion in Sobolev space. In addition, there are some results for steady MHD flows in recent years; readers interested in related issues can refer to the literature 27–29 …”