Attractors and their dimensions for the 3D Fractional Navier--Stokes--Voigt Equations
Alexei Ilyin, Varga Kalantarov, Sergey Zelik
TL;DR
The paper analyzes the long-time dynamics of the 3D fractional Navier–Stokes–Voigt equations by establishing a functional-analytic framework around the Stokes operator $A$ and its fractional powers, proving existence of a global attractor with dissipative dynamics. It then derives explicit upper bounds on the attractor's fractal dimension $\dim_f(\mathcal{A})$ using the volume contraction method in combination with advanced spectral inequalities (Berezin–Li–Yau, Lieb–Thirring, CLR), expressed in terms of the Grashof number $G$ and the dimensionless group $G_1$ for $s\in[\tfrac12,1]$. The results are regime-dependent, with stronger scaling in the high-regularization regime ($s\in[\tfrac34,1]$) and a specialized small-$G_1$ bound for the case $s=1$, improving previous non-fractional NSV estimates. The work connects the fractional regularization parameter to the attractor’s complexity, informing both analytical understanding and numerical simulations of regularized 3D flows.
Abstract
We study the dimensions of the attractors for the fractional Navier--Stokes--Voigt equations. These equations, which include a fractional order of the Stokes operator applied to the time derivative, serve as natural extensions and regularizations of the classical Navier--Stokes equations. We give a comprehensive analysis of the upper bounds for the fractal dimensions of the attractor in terms of the relevant physical parameters based on the advanced spectral inequalities such as Lieb--Thirring and Cwikel--Lieb--Rosenblum inequalities. These results extend previous works on the classical Navier--Stokes--Voigt system to the fractional setting and give an essential improvement of the estimates known before for the non-fractional case as well.