Inviscid-limit convergence conjecture for steady Navier–Stokes flows

Let $M\subset\mathbb{R}^2$ be any simply connected domain such that the constant-vorticity Euler solution $u_*$ on $M$ has a single eddy, meaning that its streamfunction has a single non-degenerate critical point. Suppose that Navier–Stokes is forced by a slip of the form specified in the paper, for example

$$f=u_*\cdot\tau+\varepsilon g$$

for some smooth function $g:\partial M\to\mathbb{R}$. Then there exists $\varepsilon_*:=\varepsilon_*(M,g)$ such that, for all $\varepsilon<\varepsilon_*$, there is weak convergence in $L^2(M)$ along a sequence of steady Navier–Stokes solutions. Inviscid-limit convergence conjecture.

$$u\to0,\qquad u^\nu\rightharpoonup\omega_0u_*$$

Here $\omega_0:=\omega_0(M,g)$ is the constant selected by the paper's boundary-layer theorem. This conjecture proposes that the nonlinearly selected constant vorticity determines the leading-order inviscid limit for general simply connected domains, extending the result known in the disk case; the statement remains unresolved in the supplied source.