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

TL;DR

This paper proves the existence, regularity, and boundary behavior of solutions to the Leray-{ alpha} model for turbulent flows with Navier slip boundary conditions, connecting it to classical Navier-Stokes solutions as parameters vary.

Contribution

It establishes the existence and regularity of solutions to the Leray-{ alpha} model with Navier slip conditions and analyzes their convergence to classical solutions under parameter limits.

Findings

Existence and regularity of weak solutions for the Leray-{ alpha} model.

Convergence of solutions to Navier-Stokes equations as filter coefficient tends to zero.

Recovery of Leray-{ alpha} solutions with Dirichlet boundary conditions as parameter approaches 1.

Abstract

In this paper, we establish the existence and the regularity of a unique weak solution to turbulent flows in a bounded domain ${\Omega}\subset \mathbb R^3$ 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.