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Analysis of the Leray-α 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 $Ω\subset \mathbb R^3$ governed by the so-called Leray-α model. We consider the Navier slip boundary conditions for the velocity. Furthermore, we show that, when the filter coefficient α 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 λ tends to 1- we recover a solution to the Leray-α model with the homogeneous Dirichlet boundary conditions.