On the classical solution for the steady triple-deck equations

TL;DR

This paper proves the existence and uniqueness of classical solutions to the steady Triple-Deck equations, which model high Reynolds number boundary layer flows over rough surfaces, using novel analytical techniques.

Contribution

It introduces a new decomposition, Green's function with Airy functions, and weighted Sobolev norms to analyze the steady Triple-Deck equations, addressing classical singularities.

Findings

Existence and uniqueness of solutions established.

A new Green's function approach overcomes singularities.

Weighted Sobolev norms provide uniform estimates.

Abstract

This paper establishes the existence and uniqueness of classical solutions to the steady Triple-Deck equations, which describe incompressible boundary layer flow over localized roughness at high Reynolds numbers. The triple-deck theory was developed to overcome the Goldstein singularity in classical Prandtl boundary layer theory, capturing the interaction between the viscous sublayer, main layer, and upper layer when surface roughness of height $O({\rm Re}^{-\frac58})$ is present. The key ingredients of this paper include: (1) A decomposition separating roughness effects from nonlinear difficulties; (2) A novel Green's function using Airy functions that overcomes low-frequency singularities via the non-vanishing property of $\sqrt{3}Ai(z)+Bi(z)$; (3) The introduction of weighted Sobolev norms $\norm{|\p_x|^{\frac{1}{18}}y^{\frac16}\omega}_{L^2}$ of the vorticity yielding $M$-independent estimates for displacement $A$. As a byproduct, local uniqueness of Couette flow is established when $F=0$.