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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.
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This content will become publicly available on August 1, 2026
Fractional Voigt-Regularization of the 3D Navier–Stokes and Euler Equations: Global Well-Posedness and Limiting Behavior
The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power $$r$$ in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power $$r \geq \frac{1}{2}$$ and for fEV when $$r>\frac{5}{6}$$. Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter $$\alpha \to 0$$. Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity $$\nu \to 0$$ as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both $$\alpha, \nu \to 0$$. Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.
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- Award ID(s):
- 2206741
- PAR ID:
- 10656976
- Publisher / Repository:
- Journal of Mathematical Fluid Mechanics
- Date Published:
- Journal Name:
- Journal of Mathematical Fluid Mechanics
- Volume:
- 27
- Issue:
- 3
- ISSN:
- 1422-6928
- Subject(s) / Keyword(s):
- Turbulence Modeling, $\alpha$-models of turbulence, Euler-Voigt, Navier-Stokes-Voigt, Fractional Derivatives, Global existence, Blow-up criteria, Voigt-regularization
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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