Regularity of solutions of the Navier-Stokes-αβ equations with wall-eddy boundary conditions

2025/12/27 by Rotundo, Nella, Tsogtgerel, Gantumur
#35Q30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.22436

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.

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