Boundary layer analysis of the Navier-Stokes equations with Generalized Navier boundary conditions

TL;DR

This paper analyzes the boundary layer behavior of Navier-Stokes equations with generalized boundary conditions, providing explicit convergence rates to Euler solutions as viscosity vanishes, and simplifies previous approaches.

Contribution

It introduces a simple explicit corrector for boundary layer analysis under generalized Navier boundary conditions, improving convergence results and rates compared to prior work.

Findings

Convergence of Navier-Stokes to Euler solutions with explicit rates

Explicit corrector simplifies boundary layer analysis

Improved uniform convergence results in time and space

Abstract

We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $\epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $\epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $\epsilon^{3/8 - \delta}$ near the boundary and $\epsilon^{3/4 - \delta'}$ in the interior, where $\delta, \delta'$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.