Existence and regularity of solutions to the Leray-\(\alpha\) model with Navier slip boundary conditions (Q2826970)

The Leray-\(\alpha\) model in a bounded domain with \(C^{\infty}\) boundary is studied. On the boundary of the domain, Navier slip conditions for the velocity are considered. The first result concerns the existence and the uniqueness of the weak solution to the considered problem. Then a maximal regularity result is obtained. It is proven that when the filter coeffcient \(\alpha\) tends to zero, the sequence of weak solutions, \(\{(\mathbf{v}^{\alpha}, p^{\alpha})\}_{\alpha}\) converges to a suitable weak solution for the incompressible Navier-Stokes equations with Navier boundary conditions. The relation between the Leray-\(\alpha\) model and the Navier-Stokes equations with homogeneous Dirichlet boundary conditions is also analyzed. This relation is established by making the other parameter of the problem, \(\lambda\), tend to 1.