Fractional Voigt-regularization of the 3D Navier--Stokes and Euler equations: Global well-posedness and limiting behavior

TL;DR

This paper introduces a fractional Voigt-regularization technique for 3D Navier-Stokes and Euler equations, establishing global well-posedness, convergence properties, and criteria for finite-time blow-up, advancing turbulence modeling and mathematical understanding.

Contribution

It generalizes the Voigt regularization by incorporating fractional powers, proving well-posedness and convergence results, and providing blow-up criteria for the fractional Navier-Stokes and Euler equations.

Findings

Global well-posedness for r ≥ 1/2 in fNSV and r > 5/6 in fEV.

Solutions of fractional models converge to original equations as regularization parameters vanish.

Derived blow-up criteria based on the fractional regularization approach.

Abstract

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.