On a 3D Stokes eigenvalue problem under Navier slip-with-friction boundary conditions and applications to Navier-Stokes equations

TL;DR

This paper analyzes the spectral properties of a 3D Stokes eigenvalue problem with Navier slip-with-friction boundary conditions in a simple geometry, and applies findings to establish conditions for global strong solutions of the Navier-Stokes equations.

Contribution

It provides an explicit spectral analysis of the 3D Stokes problem with Navier boundary conditions and uses this to identify data classes that ensure global strong solutions for Navier-Stokes.

Findings

Explicit eigenvalues and eigenfunctions characterized

Identification of data classes leading to global solutions

Spectral analysis in a simplified geometric setting

Abstract

In this paper we consider, by means of a precise spectral analysis, the 3D Navier-Stokes equations endowed with Navier slip-with-friction boundary conditions. We study the problem in a very simple geometric situation as the region between two parallel planes, with periodicity along the two planes. This setting, which is often used in the theory of boundary layers, requires some special treatment for what concerns the functional setting and allows us to characterize in a rather explicit manner eigenvalues and eigenfunctions of the associated Stokes problem. These, will be then used in order to identify infinite dimensional classes of data leading to global strong solutions for the corresponding evolution Navier-Stokes equations.