Global well-posedness of the velocity–vorticity-Voigt model of 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.∇ · u = 0, u(·, 0) = u 0 and w(·, 0) = w 0 .where u = (u 1 , u 2 , u 3 ) represents an averaged velocity, w = (w 1 , w 2 , w 3 ), which plays the role of vorticity but for which we do not assume w = ∇ × u, and f is an external forcing term. Without loss of generality, we assume for our analysis in later sections that the viscosity ν = 1. Note that in the case where α = 0, the system formally reduces the velocity-vorticty formulation, while for α > 0, if one imposes w = ∇ × u, the system formally reduces to the Navier-Stokes-Voigt equations.The term −α 2 ∆∂ t u in (1.3a) is often referred to as the "Voigt-term", due to an application of modeling Kelvin-Voigt fluids by A.P. Oskolkov [49,50] (see also [28]). In the context of the velocity formulation use of the Voigt term was first proposed as a regularization for either the Navier-Stokes (for ν > 0) or Euler (for ν = 0) equations in [4], for small values of the regularization parameter α. This paper also proved global well-posedness of the Voigt-regularized versions of the 3D Euler and 3D Navier-Stokes equations.These equations have been studied analytically and extended in a wide variety of contexts (see, e.g.], and the references therein). Voigtregularizations of parabolic equations are a special case of pseudoparabolic equations, that is, equations of the form M u t + N u = f , where M and N are (possibly non-linear, or even non-local) operators. For more about pseudoparabolic equations, see, e.g., [15,52,58,59,57,5,55,56,3].

Existence and long‐time behavior of solutions to the velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping and memory

“…By similar arguments as in Larios et al, 3 we have $$$$ {\displaystyle \begin{array}{ll}& \left|{\int}_{\tau}^T\left\langle {w}_n\times {u}_n,{P}_n\psi \right\rangle dt-{\int}_{\tau}^T\Big\langle w\times u,\psi \Big\rangle dt\right|\to 0\kern0.70em \mathrm{as}\kern0.70em n\to \infty, \\ {}& \left|{\int}_{\tau}^T\Big\langle B\left({u}_n,{w}_n\right),{P}_n\psi dt-{\int}_{\tau}^T\left\langle B\right(u,w\Big),\psi \Big\rangle dt\right|\to 0\kern0.70em \mathrm{as}\kern0.70em n\to \infty, \\ {}& \left|{\int}_{\tau}^T\Big\langle B\left({w}_n,{u}_n\right),{P}_n\psi dt-{\int}_{\tau}^T\left\langle B\right(w,u\Big),\psi \Big\rangle dt\right|\to 0\kern0.70em \mathrm{as}\kern0.70em n\to \infty .\end{array}} $$$$ …”

“…And then the trilinear form b(•, •, •) satisfies the following equalities and inequalities (see, e.g., Larios et al 3 ): For every u, v, w ∈ V, we have…”

“…In 2020, Yue and Wang 4 also considered VVV model as in Larios et al, 3 but added the damping term 𝜆w to the second equation, which parameterizes the extra dissipation occurring in the planetary boundary layer (see Zelik 5 ), such as…”

“…In 2019, Larios et al 3 pointed out that the Voigt–regularization and the velocity–vorticity formulation have not been able to overcome all the analytical and computational difficulty inherent in the 3D Navier–Stokes (NS) equations of incompressible fluid flow. Therefore, they combine these two approaches and contructed the new system which will retain the best qualities of both systems and have solutions that are closer to the actual physics of fluids, while still having enough regularization that the equations are better behaved from the standpoints of mathematical analysis, numerical stability, and computational efficiency.…”

“…The new system is a new regularization of the 3D NS equations, which is called the 3D VVV model, with a Voigt regularization term added to momentum equation in velocity‐vorticity form, but with no regularizing term in the vorticity equation. Larios et al 3 only proved the global well‐posedness by Galerkin approximation and convergence properties of the system.…”