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

TL;DR

This paper analyzes the fractal dimensions of attractors in the fractional Navier--Stokes--Voigt equations, extending classical results to fractional operators and providing improved upper bounds based on spectral inequalities.

Contribution

It extends previous work on Navier--Stokes--Voigt systems to fractional cases and improves the estimates of attractor dimensions using advanced spectral inequalities.

Findings

Derived upper bounds for attractor dimensions in fractional Navier--Stokes--Voigt equations.

Extended classical results to fractional operators with improved estimates.

Utilized spectral inequalities like Lieb--Thirring and Cwikel--Lieb--Rosenblum.

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.