The Asymptotic Behavior For A Class Of Global Modified Navier-Stokes Equations

In this paper,we consider the global modified Navier-Stokes system with frac-tional dissipation term A2?u and FN(???u?),which were firstly introduced by Cara-ballo,Kloeden and Real(Adv.Nonlinear Stud.6:411-436,2006),see[2],(?)This study shows that the system admit at least one global weak solution when 4?+2?>5,?>1/2.Particularly,when ?>?,if 4?~2-5?+ 2?2 ? 0 or 2? + 4?>5 the weak solution is unique.And there exist a unique global strong solution when 4? + 2?>5,?>1/2,s??.The existence of the global attractor for the global strong solution in H1 can be proved.Moreover,if s?max {1,?},f ?Hs0,s0= s-1+?,we can give a finte fractal dimension upper bound for the attractor.This paper is divided into four parts.In chapter 1,we give a introduction of dynamics system which include some definitions and lemmas.In chapter 2,we provide some basic knowledge,which includes some functions space,inequalities,lemmas.In chapter 3,we give the existence and unique of the solution for the global modified Navier-Stokes system.In chapter 4,the existence of global attractor A is obtained and the upper bound for it's fractal dimension is provided under suitable conditions.