On the three dimensional generalized Navier-Stokes equations with damping

Abstract

In this paper, we consider the long time behavior of solutions of the three dimensional (3D) generalized Navier-Stokes equations with damping. This family of 3D generalized Navier-Stokes equations with damping can be viewed as an interpolation model between subcritical (if \(\alpha >\frac{5}{4}\)), critical (if \(\alpha =\frac{5}{4}\)), and supercritical dissipations (if \(\alpha <\frac{5}{4}\)) and it may reduce to many models by varying the parameters. First, in a periodic bounded domain, we study the existence and uniqueness of weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since our system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu called the evolutionary system to obtain various attractors and their properties. Moreover, the determining wavenumbers are also investigated here and this is the first result for a fractional equation.