Vorticity Leray-\(\alpha \) model for Navier–Stokes equations with viscosity depending on the distance to the wall

We introduce a vorticity Leray- \(\alpha \) $\alpha$ model with eddy viscosity depending on \(d(x,\partial \Omega )^\eta \) $d{(x,\partial \mathrm{\Omega })}^{\eta }$ where \(\partial \Omega \) $\partial \mathrm{\Omega }$ is the boundary of the domain and \(\eta \in ]0;1[\) $\eta \in ]0;1[$ . We prove that this system admits fairly regular weak solutions converging when \(\alpha \) $\alpha$ goes to 0 to the solution of a reference system