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

Nella Rotundo; Gantumur Tsogtgerel

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arXiv · 2512.22436

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

Nella Rotundo·Gantumur Tsogtgerel

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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.

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Nella Rotundo, Gantumur Tsogtgerel. 2025-12-27. Regularity of solutions of the Navier-Stokes-{\alpha}{\beta} equations with wall-eddy boundary conditions. https://arxiv.org/abs/2512.22436

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