Gevrey regularity of the global attractor of the 3D Navier-Stokes-Voight equations

TL;DR

This paper proves that the global attractor of the 3D Navier-Stokes-Voight equations consists of analytic functions, indicating exponential decay in the spectrum of solutions and suggesting similar statistical properties to the classical 3D Navier-Stokes equations.

Contribution

It establishes the analyticity of the global attractor for the 3D Navier-Stokes-Voight model and provides bounds for the exponential decay scale of solutions.

Findings

Global attractor consists of analytic functions.

Solutions exhibit exponentially decaying spectral tails.

Bounds for dissipation length scale match those of 3D Navier-Stokes.

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 $D 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.