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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 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