Chapter 1 Preliminary
In this chapter， we shall recall some basic knowledge in functional analysis (harmonic analysis) and idds for nonlinear evolutionary equations， most of which will be used in the subsequent chapters. The reader can easily find the detailed proofs in the related literature， see， e.g.， Adams [1]， Babin and Vishik [5]，Chemin [20]，Chepyzhov and Vishik [24]， Constantin and Foias [27]，Evans [30]，Hale [54]， Hille and Phillips [57]， Kato [93]，Ladyzhenskaya [74，75]，Lemarie-Rieusse [76]， Lions [78]， Liu and Zheng [81]， Lorentz [82]，Liu and Zheng [80]，Maz，ja [89]，Miao [90， 91]， Nirenberg [97]， Novotny and Strauskraba [98]， Pazy [104]， Qin [112]， Robinson [125]， Rudin [126]， Sell and You [129]， Serrin [130]， Smoller [135]， Sobolev [136]， Sogge [137]，Sohr [138]， Stein [141]， Temam， Babin and Vishik [5]，Sell and You [129]， Temam [144-146]， Triebel [147，148]， Walter [150]， Yosida [155]，Zheng [156， 157]， Zhongj Fan and Chen [161]，etc.
1.1 Some Useful Inequalities
In this section， we shall recall some inequalities which will be used in the subsequent chapters.
Throughout next chapters， we set， C will stand for a generic positive constant， depending on Q and some constants， but independent of the choice of the initial time and t. We introduce the Hausdorff semi-distance in X between two sets and.
We set， divu， H is the closure of the set E infe topology， V is the closure of the set E in topology， W is the closure of the set E in (H2(Q))k topology， i.e.，
(1.1.1)
(1.1.2)
P is the Helmholz-Leray orthogonal projection in onto the space is the Stokes operator subject to the nonslip homogeneous Dirichlet boundary condition with the domain，and A is a self-adjoint positively defined operator on H. is a compact operator from H to H. The sequence {cOj}^ is an orthonormal system of eigenfunctions of are the eigenvalues of the Stokes operator A corresponding to the eigenfunctions Let
(1-1.3)
where is a Hilbert space， and. Clearly， Vo = H， and ，Hf and are dual spaces of H and V respectively， where the injection is dense， continuous. denote the norm and inner product of H， respectively， i.e.，
(1.1.4)
and denote the norm and inner product in V， respectively， i.e.，
(1.1.5)
and
(1.1.6)
The norm denotes the norm in denotes the dual product in V and Vf. We define the following bilinear form operator:
(1.1.7)
and the trilinear form operator
(1.1.8)
dxi
and
(1.1.9)
where A is defined as， for all.
Clearly， the trilinear operator satisfies
(1.1.10)
(1.1.11)
(1.1.12)
(1.1.13)
(1.1.14)
Here， if the Hq norm and Hq norm replace V norm and W norm， respectively， the above inequalities also hold.
There exists a positive constant C depending only on Q such that.
Theorem 1.1.1 (Young，s Inequality). The following inequalities hold
pecially，
Theorem 1.1.2 (The Cauchy-Schwarz Inequality). There holds that， for all x G Rn.
(1.1.15)
(1.1.16)
(1.1.17)
(1.1.18)
Theorem 1.1.3 (Holder Inequality). Let QC]Rn be a domain，assume that and.
(1.1.19)
Theorem 1.1.4 (Minkowski5s Inequality). Assume cxd. Then for any，
(1.1.20)
Attractors for Nonlinear Autonomous Dynamical Systems
Theorem 1.1.5 (Jensen，s Inequality with Integration). Let g(x) be a function defined on (a， b) and a < g(x) < P， where a， 6， a， P are bounded constants or can reach to +00，assume f(x) is a continuous convex function defined on and ， then
(1.1.21)
Theorem 1.1.6 (Poincar6，s Inequality). Assume that QcM.n is a bounded smooth domain，then there holds for all，(1.1.22)
where is the first eigenvalue of A under the homogeneous Dirichlet boundary condition，is the norm of.
Theorem 1.1.7 (Gronwall，s Inequality). Let ， Assume satisfies
(1.1.23)
Then for ， we have
(1.1.24)
Theorem 1.1.8 (The Uniform Bellman-Gronwall Inequality). Let g(t)， h(t) and y(t) be three nonnegative locally integrable functions on such that i/(t) is locally integrable on (0， + oo) and the following inequalities are satisfied
where r， are nonnegative constants. Then we have，
Theorem 1.1.9 (Gronwall5s Inequality). Let E satisfy，
for some P > 0 and m> 0. Let now be given. Suppose that the map is continuously differentiable and fulfills the differential inequality for some e > 0 and k > 0.
Theorem 1.1.10 (Gronwall，s Inequality)， Let be an absolutely continuous function satisfying at where e > 0，，for all t>s>0 and some m > 0.
Under assumptions of theorem 1.1.6， the following inequalities hold for dimension n = 3.
Theorem 1.1.11 (Ladyzhenskaya、Inequality).
(1.1.25)
(1.1.26)
Theorem 1.1.12 (Sobolev’s Inequality). Assume that Q Ca bounded smooth domain，then for dimension n = 3， there holds
(1.1.27)
Theorem 1.1.13 (The Gagliardo-Nirenberg Inequality).
(1.1.28)
(1.1.29)
The
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