Existences Of Random Attractors For Some Evolution Equations With White Noises

In recent years, many nonlinear partial differential equations are established among the fields of Physics, Mechanics, Chemistry, Biology, Engineer, Astron-omy, Medical science, Economics and Finance。The infinite dynamic systems defined by evolution equations is an important domain of studying of partial dif-ferential equations. Its main target is to investigate the long time behavior of the dynamic system, especially, to characterize the limit of the system as time tends to infinite. The introduction of attractors is a well way to understand the asymp-totic behaviors of a dynamical system generated by nonlinear evolution equations, so that they well reflect a lot of nonlinear phenomenon in nature world.In order to consider all kinds of dynamics with stochastic influence from nature or man-made complex systems, Crauel and Flandoli[37] and Schmalfuβ[39] introduced the notion of random attractors which is a generalization of global attractors involving deterministic dynamic system. Because random attractors for the random dynamic system have strong practical background and grasp the intrinsic property of many nonlinear phenomenon, so far there are many studies on the existences of random attractors, see references [37-39,41-65] and so on. On account of the nonlinearity of equations and the complexity of stochastic noises, there are much challenges and difficulties to obtain the existence of random attractors in concrete random dynamic systems.In this paper, we study several nonlinear partial differential equations with white noises, that is, Reaction-Diffusion equation, p-Laplacian-type equation, a viscous and conductive Magneto-Hydrodynamics equations and a disperse wave Camassa-Holm equation。In the paper, we introduce the general concepts of ran-dom attractors and present sufficient and necessary conditions on the existences of random attractors in Lp(p>2) for the random dynamical systems defined on unbounded domains. We also consider the existences of random attractors in different Sobolev spaces for the random dynamical systems (RDS) generated by the above stochastic differential equations, respectively.In the second chapter, first, we consider the existence standards of random attractors in Lp(p>2) for a RDS defined on unbounded domains, and investigate the relationship between the random attractors in L2and Lp. We get the following two results.Theorem2.1:Let φ be a continuous RDS on L2and a RDS on Lp, where2≤p<∞. Assume that φ has an (L2, L2)-random attractor. Then φ has an (L2, Lp)-random attractor if and only if the following conditions hold:(ⅰ) φ has an (L2, Lp)-random absorbing set{K0(ω)}ω∈Ω;(ⅱ) φ is (L2,Lp)-asymptotically compact.Moreover, the (L2, L2)-random attractor and the (L2, Lp)-random attractor are identical in the set inclusion-relation sense.Theorem2.2:Let φ be a continuous RDS on L2and a RDS on Lp, where2≤p≤∞. Assume that φ has an (L2, L2)-random attractor. Then φ has an (L2,Lp)-random attractor if and only if the following conditions hold:(ⅰ) φ has an (L2, Lp)-random absorbing set{K0(ω)}ω∈Ω;(ⅱ) For any ε>0and every{B(ω)}ω∈Ω∈D, there exist positive constants M=M(ε, B, ω) and T=T{ε, B,ω) such that for all t≥T, where Φ(t)=φ(t,θ-tω).Moreover, the (L2,L2)-random attractor and the (L2, Lp)-random attractor are identical in the set inclusion-relation sense.Second, as an application of our theoretical results we discuss the Reaction-Diffusion equations with additive noise on the whole space RN, that is, the following equation: with initial value condition u(x,0)=u0(x), x∈RN, where λ is a positive constant. The unknown u=u(x, t) is a real valued function of x∈RN and t≥0. g and hj(1≤j≤m) are given functions on RN. The non-linear function f(x, u) satisfies the following growth and dissipative conditions: for x∈RN,u∈R, where α1,α2and β are positive constants and p≥2.{ωj(t)}j=1m are independent two-side real-valued Wiener processes on a complete probability space (Ω,F, P), where Ω={ω∈C(R, Rm):ω(0)=0}, F is the Borel σ-algebra induced by the compact-open topology of Ω and P is the corresponding Wiener measure on (Ω,F).By checking that the tail of solution vanishes as time tends to infinite, we obtain the asymptotic compactness in Lp for the RDS generated by (1), which is called as the (L2, Lp)-asymptotically compact for this RDS. But [61] proved the existence of the (L2, L2)-random attractor for the same RDS, so we haveTheorem2.8: Assume that g∈L2and (a1)-(d1) hold. Then the RDS φ generated by the initial problem (1) has an unique (L2, Lp)-random attractor {Ap(ω)}ω∈Ω, which is a compact and invariant tempered random subset of Lp attracting every tempered random subset of L2. Furthermore, Ap(ω)=A(ω), where {A(ω)}ω∈Ω is the (L2, L2)-random attractor.In the third chapter, we consider the stochastic p-Laplacian-type equation with additive noise defined on bounded domains D(?)RN, which reads with boundary conditions and initial value conditions whereΦp(s)=|s|p-2s, p≥2; The exterior forced function g(x,s) defined in D×R is subjected to the following growth and monotonicity assumptions where2≤q≤p<∞.φj∈W04,p(D) for j=1,2,...,m.{Wj(t)}j=1m are inde-pendent two-side real-valued Wiener processes on a complete probability space (Ω,F,P), where Ω={ω∈C(R,Rm):ω(0)=0}, F is the Borel a-algebra induced by the compact-open topology of Ω and P is the corresponding Wiener measure on (Ω,F).Suppose that the assumptions (a2)-(c2) hold, we show that the unique existence of the solution to equation (2) and continuous dependence of the so-lution on initial values in L2, and therefore we obtain the unique existence of a continuous RDS on L2space. By asymptotic a priori estimate and using Dirichlet forms of Laplacian, we obtain the regularity of solution to equation (2). Then the existence of a compact absorbing set is established in L2(D) for the RDS generated by equation (2). So we haveTheorem3.5:Assumed that g satisfies (a2)-(c2) and f is given in V'. Then the RDS φ generated by the stochastic equation (2) possesses a random attractor {A(ω)}ω∈Ω defined as where B(L2) denotes all the bounded subsets of L2(D) and the closure is the L2(D)-norm. Theorem3.6: Assume that g satisfies (a2)-(c2) and f is given in V', C3>0. Then the RDS φ generated by the solution to (2) possesses a single point attractor {A(ω)}ω∈Ω in L2(D), i. e. there exists a single point ξ0(ω) in L2(D) such that A(ω)={ξ0(ω)}.In the forth chapter, we consider a viscous Magneto-Hydrodynamics equa-tions with additive noise on bounded and smooth domains O(?)R2, which reads with boundary conditions: on (0,+∞)×T, v=0; B.n=0, curl B=0, and initial value conditions v(x,0)=v0, B(x,0)=B0, x∈O, where P=P(x,t) is the unknown total pressure, v1,v2>0. In equations (3), v=(v1(x,t),v2(x,t)),B=(B1(x,t),B2{x,t)), G(x)=(g1(x),g2(x))∈L2(O)2.{ωj(t)}j=12are independent two-sided real-valued Wiener processes on a prob-ability space (Ω,F,P), where Ω={ω∈C(R,R2): ω(0)=0}, F is the Borel σ-algebra induced by the compact-open topology of Ω and P is the cor-responding Wiener measure on (Ω,F). The nonlinear vector function F(x,v)=(f1(x,v1), f2{x, v2)) satisfies the following conditions: x∈(?),v∈R2, where|v|is the modular of the vector v∈R2, and α1,α2,C1are positive con-stants. We have the following result. Theorem4.4:Assume that (a3)-(d3) hold and G∈L2(O)2. Then the RDS φ generated by stochastic MHD equations (3) possesses an unique random attractor {A(ω)}ω∈Ω in L2(O)2×L2(O)2.{A(ω)}ω∈Ω is an invariant and compact random set which attracts every tempered random sets of L2(O)2×L2(O)2.In the last chapter, we consider a dispersive wave model, that is the dissi-pative Camassa-Holm equation with additive noise with periodic boundary conditions. In (4), W(t) is independent two-sided real-valued Wiener process on a probability space (Ω,.F, P). Q:Rm→L2(I) is a bounded linear operator.We show that the RDS generated by equation (4) possesses random absorb-ing sets in Hilbert space H1(Ⅰ) and H2(Ⅰ), respectively. Then by embedding theorems we getTheorem5.5:Assume that ε>0. Then the RDS φ generated by Camassa-Holm equation (4) possesses a random attractor{A(ω)ω∈Ω in H2(Ⅰ), which attracts every deterministic bounded subset of H1(Ⅰ).