| The attractor is one of the most important problems recently.The concept of global attractor has become a very useful tool to describe the long-time behavior of the dynamical systems generated by certain differential equations.The global attractor is an invariant set that attracts all the trajectories of the system.Its geometry can be very complicated and rellects the compliexity of long-time dynamics of the systems. The deterministic cases of many equations have been studied systematically by others.In 1994,Crauel H. and Flandoli F. [2]have defined random attractors for random dynamical systems by defining an attracting set as a set that attracts any orbit starting from -∞. In the stochastic case,the global attractor is a compact random invariant set which attracts all deterministic bounded sets. So,theories of attractor have obtained more development.In this paper,we consider the random attractors of Cahn-Hilliard equations.This equation has been introduced by Cahn and Hilliard [7] to describe the evolution of the phase separation of a binary alloy the temperature has been quenched.The deterministic case has been studied systematically by many authors .However,in many practical circumstances,small irregularity has to be taken account.Thus,it is necessary to add to the equation a random force,which is in general a space-time white noise,as considered recently by many authors for other equations.We recpetively add an additive noise and a multiplicative noise with Cahn-Hilliard equation.We'll consider the two following stochastic Cahn-Hilliard equations perturbed respectively by an additive white noise and a multiplicative noise forms:du + (â–³2u -â–³f(u))dt=φ)dw, (x, t)∈G×R. (0.1)anddu + (â–³2u -â–³f(u))dt = u o dw, (x, t)∈G×R. (0.2)here (x,t)∈G×R, G =multiply from i=1 to n(0,Li), Li > 0,is an open bounded set of Rn,n = 1,2 or 3,with a smooth boundaryГ. f is a polynomial of order 2p - 1f(u)=sum from j=1 to (2p-1)ajuj,p∈N,p≥2,a2p-1>. (0.3) φ∈D(A2)∩(?)2(G),where Aφ∈Z,A, D(A2), (?)2(G) defined in this paper. The two equations are associated with boundary conditions which could be the periodic boundary conditionThe white noise described by a process w(t) results from the fact small regularity has to be taken account in some circumstance. Here,we assume that w(t) is a two sided Wiener processes on the probability space (Ω,F, P),whereΩ= {ω∈C(R,R) :ω(0) = 0}. F is the Borel sigma-algebra induced by the compact-open topology ofΩ, P is a Wiener measure.Da Prato G. and Debussche A.[4] have proved the existence of solution for (0.1).Similiarly ,we can obtain the existence of solution for (0.2). Of course, the unique solution will generate a random dynamical system(RDS)S.Our main aim in this paper is to prove the existence of random attractors with respect to this RDS.Since the two equations and boundary conditions are special,it is difficult(or impossible) to obtain a global attractor even for the determistic case. Thus,we consider the existence of random attractors in V-1(defined in this paper).By, theorem 1.3.21,we have to get a compact random asborbing set which absorbs all determinstic bounded sets B (?) V-1.So we have to make the following work: First,proving every solution of corrosponding equation is a determinstic bounded set.Second,proving the determinstic bounded set with respect to every solution is absorbed by a compact random set.When prove the compactness of the absorbing set obtained,almost authors applied the compact embedding of two spaces.However, in this paper,it is impossible that V-1 (?) L2(·) is compact.We applied the compactness of the operator A-1/2(defined in this paper) to prove the compactness of absorbing set.In chapter 2,we prove the solution of the Cahn-Hilliard equation with additive noise in V-1 has the following property:Lemma 2.1 There exists a random radiusγ1(ω) > 0,such that for allÏ> 0 there exists (a determinstic) (?)≤-1 such that the following holds P-a.s., for all t0≤(?) and for all u0∈V-1 with‖u0‖-1≤Ï,the solution u(t,ω;t0,u0) of equation (0.1) with v(t0) = u0 - z(t0) satisfies the inequality‖u(t,ω;t0,u0)‖-12≤γ12(ω).To obtain a compact random absorbing set,we prove the following lemma.Lemma 2.2 There exists random radius c1(ω), c2(ω), c3(ω) > 0,such that for all p > 0 there exists (a determinstic) (?)≤-1 such that the following holds P-a.s., for all t0≤(?) and for all u0∈V-1 with‖u0‖-1≤Ï,the solution u(t,ω;t0,u0) of equation (0.1)and the solution v(t,ω;t0,v0) of equation (2.7) (see chapter 2) respectively satisfies∫-10‖u(t)‖12 dt≤c12(ω),∫-10 |v(t)|2dt≤c22(ω),∫-10‖u(t)‖Z2pdt≤c32(ω).The following lemma gives a compact random set absorbing every bounded set in V1.Lemma 2.3 There exists a compact random set B(0,γ2(ω)) (?) V-1,such that for allÏ> 0 there exists (a determinstic) (?)≤-1 such that the following holds P-a.s., for all t0 <(?) and for all u0∈V-1 with‖u0‖-1≤Ï,the solution u(0,ω;t0,u0) of (0.1) satisfies u(0,ω;t0,u0) (?) 5(0,γ2(ω)).Finally,we get a useful result.Theorem 2.4 The random dynamical system associated with the Cahn-Hilliard equation with additive noise (0.1) has a global attractor,which is contained in a random ball of V-1 and attracts all deterministic bounded sets of V-1.For equation (0.2),we also get the following similiar results in chapter 3.Lemma 3.1 There exists a random radiusλ1(ω) > 0,such that for allÏ> 0 there exists (a determinstic) (?)≤-1 such that the following holds P-a.s., for all t0 <(?) and for all u0 (?) V-1 with‖u0‖-1≤Ï, the solution u(t,ω;t0,u0) of equation (0.2) with v(t0) = u0 - z(t0) satisfies the inequality‖u(t,ω;t0,u0)‖-12≤λ12(ω).Lemma 3.2 There exists random radius a1(ω), a2(ω) > 0,such that for allÏ> 0 there exists (a determinstic) (?)≤-1 such that the following holds P-a.s., for all t0≤(?) and for all u0∈V-1 with‖u0‖-1≤Ï, the solution u(t,ω;t0,u0) of equation (0.2) and the solution v(t,ω;t0,v0) of (3.5) (see chapter 3) respectively satisfies∫-10‖u(t)‖12dt≤a12(ω),∫-1|v(t)|2dt≤a22(ω).Lemma 3.3 There exists a compact random set B(0,λ2(ω)) (?) V-1,such that for allÏ> 0 there exists (a determinstic) (?)≤-1 such that the following holds P-a.s., for all t0≤(?) and for all u0∈V-1 with‖u0‖-1≤Ï, the solution u(0,ω;t0,u0) of (0.1) satisfies u(0,ω;t0,u0) (?) B(0,λ2(ω)).Theorem 3.4 The random dynamical system associated with the Cahn-Hilliard equation with multiplicative noise (0.2) has a global attractor,which is contained in a random ball of V-1 and attracts all deterministic bounded sets of V-1.
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