Global well-posedness of the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations

TL;DR

This paper introduces a new regularized 3D Navier-Stokes model with a velocity-vorticity-Voigt formulation, proving its global well-posedness, convergence to classical solutions, and providing a blow-up criterion based on the regularization.

Contribution

It proposes the 3D velocity-vorticity-Voigt model with a novel regularization, establishing its mathematical well-posedness and convergence properties.

Findings

Proved global existence and regularity of the VVV model.

Showed convergence of the model's velocity and vorticity to Navier-Stokes solutions as the regularization parameter tends to zero.

Provided a finite-time blow-up criterion for the 3D Navier-Stokes equations.

Abstract

The velocity-vorticity formulation of the 3D Navier-Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier-Stokes equations, which we call the 3D velocity-vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity-vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier-Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier-Stokes equations based on this inviscid regularization.