Regularity of solutions of the Navier-Stokes-{\alpha}{\beta} equations with wall-eddy boundary conditions

TL;DR

This paper proves global well-posedness and regularity for the Navier-Stokes-αβ equations with wall-eddy boundary conditions, modeling near-wall turbulence, and provides the first complete analytical treatment of this model.

Contribution

It introduces a rigorous mathematical analysis of the Navier-Stokes-αβ system with wall-eddy boundary conditions, establishing well-posedness and regularity results.

Findings

Proved symmetry and Gårding inequality for the stationary system

Verified ellipticity and boundary conditions for regularity

Established global regularity, uniqueness, and stability for the nonlinear evolution

Abstract

We establish global well-posedness and regularity for the Navier-Stokes-{\alpha}{\beta} system endowed with the wall-eddy boundary conditions proposed by Fried and Gurtin (2008). These conditions introduce a tangential vorticity traction proportional to wall vorticity and provide a continuum-mechanical model for near-wall turbulence. Our analysis begins with a variational formulation of the stationary fourth-order system, where we prove symmetry and a G{\aa}rding inequality for the associated bilinear form. We then verify Douglis-Nirenberg ellipticity and the Lopatinskii-Shapiro covering condition, establishing full Agmon-Douglis-Nirenberg regularity for the coupled system. Building on this framework, we derive a hierarchy of energy estimates for the nonlinear evolution equation, which yields global regularity, uniqueness, and stability. To our knowledge, this provides the first complete analytical treatment of the wall-eddy boundary model of Fried and Gurtin.