Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Abstract

Recently, the Navier-Stokes-Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides an additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier-Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale -the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier-Stokes equations. Our estimate coincides with similar available lower bound for the smallest dissipation length scale of solutions of the 3D Navier-Stokes equations.MSC Classification: 35Q30, 35Q35, 35B40, 35B41, 76F20, 76F55

Singular Limits of Voigt Models in Fluid Dynamics

“…The estimates of the fractal and Hausdorff dimensions of the global attractor were later improved in [33], where also an upper bound on the number of asymptotic determining modes of the solutions was provided. Concerning the regularity of the attractor, it can be proved to be as smooth as the forcing term f permits, and even real analytic, whenever f is analytic as well [32].…”

“…Moreover, the robust analytical properties of the 3D NSV model ensure the computability of solutions and the stability of numerical schemes. Finally, there is evidence that the 3D NSV with a small regularization parameter enjoys similar statistical properties as the 3D NSE [32,44]. In this connection, understanding the long-time behavior of solutions to the 3D NSV equation is crucial.…”

Upper bound of decay rate for solutions to the Navier–Stokes–Voigt equations in R3

“…On bounded domains, the NSV equations have been deeply studied. One can see Oskolkov [16], Kalantarov [10,11], Kalantarov et al [12], Kalantarov and Titi [13], García-Luengo et al [9], Gao and Sun [8], Qin et al [18], etc.…”

On a Computational Advantageous Voigt Regularization for Geophysical Flows

“…BV-Voigt is a regularization of the BV model based on the Navier-Stokes-Voigt (NSV) equations [5], an incompressible viscoelastic model which is a smooth regularization of Navier-Stokes equations [4]. In [5] it is showed that NSV reduces the stiffness of direct numerical simulations of turbulent flows, with small impact in the energy containing scales.…”