Analysis of the Leray-{\alpha} model with Navier slip boundary condition

In this paper, we establish the existence and the regularity of a unique weak solution to turbulent flows in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ governed by the so-called Leray-{\alpha} model. We consider the Navier slip boundary conditions for the velocity. Furthermore, we show that, when the filter coefficient {\alpha} tends to zero, the weak solution constructed converges to a suitable weak solution to the incompressible Navier Stokes equations subject to the Navier boundary condition. Similarly, if {\lambda} tends to 1- we recover a solution to the Leray-{\alpha} model with the homogeneous Dirichlet boundary conditions.