Attractors and their dimensions for the 3D Fractional Navier--Stokes--Voigt Equations

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.