Regularity of solutions of the Navier-Stokes-αβ equations with wall-eddy boundary conditions
Nella Rotundo, Gantumur Tsogtgerel
TL;DR
The paper analyzes the Navier-Stokes-$\alpha\beta$ system with wall-eddy boundary conditions, proving global existence and regularity of solutions. It develops a pressure-free variational formulation for the linear stationary problem, establishes a Gårding inequality, and places the system in the Agmon–Douglis–Nirenberg framework to obtain sharp elliptic regularity for $u$ and $p$. Building a nonlinear evolution theory on top of this, the authors derive a hierarchy of energy estimates that yield global well-posedness and uniqueness, and they show a vanishing-regularization limit to Leray–Hopf solutions. This work provides the first complete analytical treatment of Fried–Gurtin’s wall-eddy boundary model, connecting near-wall turbulence boundary conditions to rigorous PDE theory with global-in-time control.
Abstract
We establish global well-posedness and regularity for the Navier-Stokes-αβ 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 Gårding 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.