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

Abstract

We establish the existence and regularity of a unique weak solution to turbulent flows in a bounded domain $\Omega\subset\mathbb R^3$ governed by the Leray-$\alpha$ model with Navier slip boundary condition for the velocity. Furthermore, we show that when the filter coefficient $\alpha$ tends to zero, these weak solutions converge to a suitable weak solution to the incompressible Navier Stokes equations subject to the Navier boundary conditions. Finally, we discuss the relation between the Leray-$\alpha$ model and the Navier-Stokes equations with homogeneous Dirichlet boundary condition.

Keywords

• Turbulence model
• existence of solutions
• weak solution