[Paper Review] Boundary layers and the vanishing viscosity limit for incompressible 2D flow

Milton C. Lopes Filho|ArXiv.org|Dec 6, 2007

Navier-Stokes equation solutions22 references9 citations

TL;DR

This paper presents a rigorous mathematical analysis of boundary layers and the vanishing viscosity limit in two-dimensional incompressible flows, focusing on the Navier-Stokes equations with no-slip and Navier friction boundary conditions. It establishes $L^p$ vorticity estimates, proves weak* convergence of vorticity to a limit involving boundary measures, and identifies acceleration of the boundary as a key driver of boundary layer strength, highlighting the critical role of vortex-sheet regularity in the inviscid limit.

ABSTRACT

This manuscript is a survey on results related to boundary layers and the vanishing viscosity limit for incompressible flow. It is the lecture notes for a 10 hour minicourse given at the Morningside Center, Academia Sinica, Beijing, PRC from 11/28 to 12/07, 2007. The main topics covered are: a derivation of Prandtl's boundary layer equation; an outline of the rigorous theory of Prandtl's equation, without proofs; Kato's criterion for the vanishing viscosity limit; the vanishing viscosity limit with Navier friction condition; rigorous boundary layer theory for the Navier friction condition and boundary layers for flows in a rotating cylinder.

Motivation & Objective

• To clarify the mathematical structure of boundary layers in the vanishing viscosity limit for incompressible 2D flows.
• To analyze the convergence of solutions to the Navier-Stokes equations to solutions of the Euler equations in bounded domains with boundaries.
• To investigate the role of boundary conditions—particularly Navier friction and no-slip—on the formation and regularity of boundary layers.
• To identify the conditions under which vorticity converges weakly* and the implications for the well-posedness of the inviscid limit.
• To highlight open problems in the theory, especially concerning vortex-sheet regularity and boundary layer separation.

Proposed method

• Derives Prandtl’s boundary layer equation via asymptotic analysis of the Navier-Stokes equations in the small viscosity limit.
• Applies Kato’s criterion to characterize the vanishing viscosity limit in bounded domains, linking convergence to the absence of boundary layer formation.
• Establishes $L^p$ vorticity estimates for the Navier friction condition, showing convergence in vorticity for $p > 1$ under $L^p$ boundary data.
• Uses geometric optics-inspired methods to rigorously analyze boundary layer expansions under the Navier condition.
• Analyzes circularly symmetric flows in a rotating cylinder to construct explicit examples of boundary layer behavior under no-slip conditions.
• Demonstrates that vorticity generated by boundary layers is proportional to the acceleration of the boundary relative to the flow.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of the Navier-Stokes equations converge to a solution of the Euler equations in a bounded domain with a boundary?
• RQ2 How does the Navier friction condition affect the well-posedness and regularity of boundary layer solutions compared to the no-slip condition?
• RQ3 What is the precise nature of vorticity convergence in the vanishing viscosity limit, particularly in the presence of boundary layers?
• RQ4 Can the boundary layer be described as a vortex sheet, and what regularity conditions are required for such a description to hold?
• RQ5 Is boundary layer separation possible under the Navier friction condition, and how does it affect the vanishing viscosity limit?

Key findings

• The vorticity $\omega^\nu$ of the Navier-Stokes solution converges weak-* to a limit $W = \omega_0 + \mu$ in $L^\infty_{\text{loc}}((0,\infty);BM(\overline{D}))$, where $\mu$ is a measure supported on the boundary, indicating vortex-sheet formation.
• For $\alpha \in BV$, the vorticity $\omega^\nu$ does not converge in $L^1$ to $\omega_0$, implying that $L^2$ velocity convergence cannot be improved to convergence in derivatives.
• The strength of the boundary layer is proportional to the acceleration of the boundary relative to the adjacent flow, suggesting acceleration as a key driver of boundary layer intensity.
• Under the Navier friction condition, $L^p$ vorticity estimates are established for $p > 1$, and convergence in $L^p$ holds under $L^p$ boundary data, extending earlier results.
• The theory of renormalized solutions of DiPerna-Lions implies that $L^p$ vorticity norms are conserved for $p \geq 2$ in the absence of boundaries, but this conservation may not hold in the presence of boundary layers.
• The paper constructs an explicit example of boundary layer behavior in a rotating cylinder, showing that boundary layer separation may occur under certain initial vorticity conditions, such as odd eigenfunctions of the Dirichlet Laplacian.