[Paper Review] Verification of Prandtl boundary layer ansatz for the steady electrically conducting fluids with a moving physical boundary

Shijin Ding, Zhilin Lin+1 more|arXiv (Cornell University)|Jan 2, 2020

Navier-Stokes equation solutions22 references4 citations

TL;DR

This paper verifies the Prandtl boundary layer ansatz for two-dimensional steady viscous incompressible magnetohydrodynamics (MHD) flows with a moving boundary, establishing the inviscid limit in Sobolev spaces under conditions of equal Reynolds numbers and non-degenerate tangential magnetic fields. The key result confirms the validity of the boundary layer expansion and provides $L^∞$ error estimates despite strong boundary layers.

ABSTRACT

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 $\{(X, Y)\in[0, L] imes\mathbb{R}_+\}$ 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 $ε$, 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 $L^\infty$ sense.

Motivation & Objective

• To establish the validity of the Prandtl boundary layer ansatz for steady viscous incompressible MHD flows with a moving flat boundary.
• To analyze the inviscid limit of the 2D MHD equations as the viscosity and resistivity coefficients vanish proportionally to a small parameter $\epsilon$.
• To determine under what conditions the boundary layer correctors accurately capture the singular behavior near the moving boundary.
• To derive rigorous error estimates in $L^\infty$ norm for the difference between the viscous solution and the inviscid limit plus boundary layer correction.

Proposed method

• Introduce Prandtl fast variables $x = X$, $y = Y/\sqrt{\epsilon}$ to rescale the boundary layer region.
• Define transformed variables for velocity and magnetic field components to isolate boundary layer dynamics, including $V^\epsilon = V/\sqrt{\epsilon}$ and $G^\epsilon = G/\sqrt{\epsilon}$.
• Derive the rescaled MHD system in the fast variables, revealing the boundary layer structure with $\nu\partial_{yy}$ and $\kappa\partial_{yy}$ terms dominating near $y=0$.
• Apply energy estimates in weighted $L^2$ spaces, exploiting decay properties of boundary layer profiles and weighted derivatives.
• Use weighted $L^\infty$ bounds on the streamwise derivatives of the boundary layer profiles to control nonlinear terms.
• Establish the key estimate $\|\partial_y v^\epsilon\|_{L^2} + \|\partial_y g^\epsilon\|_{L^2} \lesssim \|h_s/u_s\|_{L^\infty} + \|y\partial_y(u_s,h_s)\|_{L^\infty}$ under smallness assumptions.

Experimental results

Research questions

• RQ1 Under what conditions does the Prandtl boundary layer ansatz hold for steady 2D MHD with a moving boundary?
• RQ2 Can the inviscid limit of the viscous MHD equations be rigorously justified in Sobolev spaces when strong boundary layers form?
• RQ3 What role does the tangential magnetic field play in stabilizing or destabilizing the boundary layer structure?
• RQ4 How do the relative orders of viscosity and resistivity affect the validity of the boundary layer expansion?
• RQ5 Can $L^\infty$ error estimates be derived for the difference between viscous and inviscid solutions in the presence of strong boundary layers?

Key findings

• The inviscid limit of the 2D steady viscous incompressible MHD equations is established in Sobolev spaces under the assumption that hydrodynamic and magnetic Reynolds numbers are of the same order as $\epsilon^{-1}$.
• The Prandtl boundary layer ansatz is verified for the first time in the context of steady MHD with a moving boundary, even in the presence of strong boundary layers.
• The error between the viscous solution and the sum of the inviscid solution and boundary layer correctors is estimated in the $L^\infty$ norm.
• The analysis requires the tangential magnetic field to be non-degenerate near the boundary and the ratio of its strength to the tangential velocity to be suitably small.
• The key estimate is derived using weighted $L^2$ energy methods and relies on smallness of $\|h_s/u_s\|_{L^\infty}$ and $\|y\partial_y(u_s,h_s)\|_{L^\infty}$.
• The result holds despite the vorticity of the boundary layer functions becoming uncontrollable as $\epsilon \to 0$, due to careful control of nonlinear interactions through weighted norms.