Diffusion-free boundary conditions for the Navier-Stokes equations

Abstract

We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full setting of the nonlinear Navier-Stokes equation in a general bounded domain $\Omega$ of ${\mathbb{R}}^d$, $d=2$ or $3$. We show that diffusion-free boundary conditions $$\Delta u\cdot \tau {|}_{\partial \Omega }=0,u\cdot n{|}_{\partial \Omega }=0\text{ when }d=2,$$ $$\Delta u\times n{|}_{\partial \Omega }=0,u\cdot n{|}_{\partial \Omega }=0\text{ when }d=3,$$ allow for a satisfactory well-posedness theory of the full Navier-Stokes equations (global in time for $d=2$, local for $d=3$). Moreover, we perform a boundary layer analysis in the limit of vanishing viscosity $\nu \to 0$. We establish that the amplitude of the boundary layer flow is in this case of order $\nu$, i.e. much lower than in the case of standard Dirichlet or even stress-free conditions. This confirms analytically that this choice of boundary conditions may be used to reduce diffusive effects in numerical studies relying on the Navier-Stokes equation to approach nearly inviscid solutions.

31 pages, 2 figures