ASYMPTOTIC STUDY OF THE THREE-DIMENSIONAL GENERALIZED NAVIER–STOKES EQUATIONS WITH EXPONENTIAL DAMPING

We study the long time behavior of solutions of the three dimensional (3D) generalized Navier–Stokes equations with nonlinear exponential damping term a(eb|u|r−1)u (a>0,b>0,r≥1) in a periodic bounded domain. We first study the existence and uniqueness of weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. The difficult issue is that the Cauchy problem could have nonunique solution and then we cannot use directly the classical schemes. To solve this problem, we use a new framework developed by Cheskidov and Lu which called (closed) evolutionary system to obtain various attractors and its properties. Finally, we investigate the determining wavenumbers and this seems to be the first result for a fractional equation.