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

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.