Random Attractors For Generalized Kuramoto-Sivashinsky Equations With Additive Niose And Reaction-diffusion Equations With Multiplicative Niose

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 deterministic cases of many equations have been studied systematically by others.Random atractor of random dynamical systems(RDS) were first introduced by Crauel and Flandoli [2] and Schmalfuss [8], with notable develepments given in [3-9],[19-23].To prove the existence of random attractor for the RDS generated by certain stochastic partial differential equations(SPDE) ,almost all authers had relid on a basic theorem: there exists a random attractor if there is a compact absorbing random set(see[]). In this paper , to proving the random dynamical systems(RDS) generated by the the unique solution of generalized Kuramoto-Sivashinsky equation possesses a atrattor,we also rely on this theorem. However the condition may not be necessary.In[38],Li Y.R.establish a complete criterion for existence of random attractors for the RDS on a separable Banach space X.In this paper ,we'll apply this result to prove the existence of random attractors for the RDS generated by the reaction-diffusion equation with multiplicative noise in L^p(D) for p > 2.In chapter 2,we consider the equaions asdu+(D⁴u+D²u+uDu+βu)dt=sum from j=1 to mφ_jdW_j (x,t)∈I×R (1)Where D = (?)/(?)x and the internal I = (-L/2,L/2), the functionφ_j∈H⁴,j=1,2,,…,m are the time independent. The random functions W_j = W_j(t),j = 1,2…, m are independent two-side real-valued Wiener processes on a probability space (Ω,F, P),Ω= {ω∈C(R,R^m) |ω(0) = 0} The time shift is simply defined byθ_sω(t) =ω(t+s) -ω(s), t,s∈R (2) {θ_t:Ω→Ω,t∈R} are a family of measure preserving transformations such that (t,ω) (?)θ_tωis measurable,θ₀ = id, andθ_t+s =θ_tθ_s for all s,t∈R. The flowθ_t.together with the corresponding probability space (Ω, F, P,θ_t) is called a metric dynamical system.We also consider the boundary conditions asLetwith the L² scalar product and norm, denoted by (·,·), |·|. Let alsowith scalar produt and normSince the random dynamical systems(RDS) generated by the the unique solution of generalized Kuramoto-Sivashinsky equation is continuous,in order to prove the existence of the random atacttor,we first consider that the equation is coercive.If the equation is not coercive,it is difficult to obtain a general result on attractors even for the deterministic case.So,we letβ>1/4,and we get the follow results:Lemma 2.2.1 . Assumeβ> 1/4, there exists a constant K > 0 such thatLemma 2.2.2 . There exist two random radius r₁(ω),r₂(ω) > 0 satisfing the following properties: For allρ> 0, there exists t(ω)≤-1 such that, for all s≤t(ω) and all u₀∈H with |u₀| <ρthe following holds P-a.s.and an auxiliary estimateLemma 2.2.3. There exists a constant c_b > 0 such that Lemma 2.2.4. There exists a random radius r₃(ω) satisfying: For allρ> 0, there exists t(ω) < -1 such that, for all s≤t(ω) and all u₀∈H with |u₀| <ρ, the following holds P-a.s.‖u(0,ω;s,u₀)‖≤r₃(ω)where u(t,ω; s, u₀) = v(t, w; s, u₀ - z(s,ω)) + z(t,ω) .Theorem 2.2.5.Assumeβ> 1/4 andφ_j∈D(A) = (?)⁴(I), then the RDS S modelling the stochastic generalized Kuramoto-Sivashinsky equation possesses a compact random attractor A(ω) which attracts all bounded sets of H = (?)²(I).In chapter 3,we'll apply the result in [38] to prove the existence of random attractors for the RDS generated by the reaction-diffusion equation with multiplicative noise in LP(D) for p > 2.We get the follow results:Lemma 3.2.2. There exists two random variableγ₁(ω) andγ₂(ω), for any R > 0,‖u₀‖₂≤R,-1≤t≤0,there exists a s₀= s₀(R,ω) < -1,whenever s0, the following hold:‖v(t,ω,s,v(s))‖2≤γ₁(ω)‖u(t,ω;s,u₀)‖_H0¹â‰¤Î³₂(ω)Lemma 3.2.3.For every fixed p > 2, there exists a random variableγ(ω) such that for any R > 0,there exists a s′= s′(R,ω) < -1 satisfying‖u(0,ω;s,u₀)‖_p≤γ(ω) whenever s < s′,‖u₀‖₂≤RLemma 3.2.4. For anyε> 0 and bounded set B (?) L²(D), there exist s₀ = s₀(ε,B,ω)< -1,M₁ = M₁(ε,B,ω),M₂ = M₂(ε,B,ω) such that, for all s≤s₀,u₀∈B P .a.s.ω∈ΩLemma 3.2.5. For any bounded set B (?) L²(D) and P-a.s.ω∈Ω, we haveLemma 3.2.6. Let p > 0 be fixed. For every bounded set B (?) L²(D), we have Theorem 3.2.7. The RDS (?) generated by the stochastic reaction-diffusion equation (3.1) possesses a random attractor A_p(ω.) in L^p(D) for any p≥2. A_p(ω) is a invariant and compact . random set which attracts the bounded set of L²(D) under the L^p-norm topology.