Title: A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions

Authors: Gui-Qiang Chen and Zhongmin Qian

Issue: Volume 59 (2010), Issue 2, 721-760

Abstract: We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\mathbb{R}^{3}$ 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 value 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 solution of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.