A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions

TL;DR

This paper analyzes the Navier-Stokes equations with kinematic and Navier boundary conditions, developing spectral theory and proving convergence of solutions to classical boundary conditions and inviscid limits.

Contribution

It introduces a spectral framework for the Navier boundary conditions and proves the existence, convergence, and inviscid limits of solutions in this setting.

Findings

Constructed global weak solutions under Navier boundary conditions.

Showed convergence to no-slip solutions as slip length tends to zero.

Established inviscid limit results for Euler equations with slip boundary conditions.

Abstract

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\R^3$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solutions of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.