The Existence Of Exponential Attractors For Nonautonomous Navier-Stokes Equations

Recently,many authors have paid much attention to non-autonomous differential equa-tions and the processes generated by them.External forces and other time dependent lin-ear or non-linear influences in natural phenomena lead to the presence of non-autonomous terms in the associated PDE models.The solutions of a non-autonomous PDE equation strongly depend on two time variables(the final time t and the initial time s),whereas in the autonomous case they are invariant with respect to time shifts.This makes an essen-tial difference between non-autonomous and autonomous differential equations.It is well known that the long-time behavior of dissipative dynamical systems generated by evolution equations of mathematical physics can be described in terms of the so-called exponential attractor.Different approaches were made to find the counterpart of the exponential at-tractors in the non-autonomous case.Pullback exponential attractors,as a suitable notion describing long time behavior of non-autonomous dynamical systems,which is a minimal family of compact sets with finite fractal dimension semi-invariant under the process and pullback attracting each bounded subset of the phase space.In this paper we prove the existence of pullback exponential attractors for for a non-autonomous Navier-Stokes Equation in a 2D bounded domain ?:(?) Where f(t)is a given time dependent external forces field.For this end,we need the normal condition in Llov2(R;L2(?))for f(t).That is,for any ?>0,there exists ?>0 such that(?)...