Local Rigidity of the Couette Flow for the Stationary Triple-Deck Equations

TL;DR

This paper proves that the Couette flow is locally rigid within the stationary Triple-Deck equations, meaning no other stationary solutions exist nearby in a scale-invariant space, highlighting a fundamental difference from the Prandtl equations.

Contribution

It establishes the local rigidity of the Couette flow for the stationary Triple-Deck equations, addressing a longstanding open problem and contrasting with the Prandtl model.

Findings

No other stationary solutions near Couette flow in the scale-invariant space

First rigidity result for the stationary Triple-Deck equations

Contrasts with the behavior of the stationary Prandtl equations

Abstract

The Triple-Deck equations are a classical boundary layer model which describes the asymptotics of a viscous flow near the separation point, and the Couette flow is an exact stationary solution to the Triple-Deck equations. In this paper we prove the local rigidity of the Couette flow in the sense that there are no other stationary solutions near the Couette flow in a scale invariant space. This provides a stark contrast to the well-studied stationary Prandtl counterpart, and in particular offers a first result towards the rigidity question raised by R. E. Meyer in 1983.