Stachastic Boussinesq Equations With Additive Noise And It's Global Attractors

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.In 1994,H.Cranel and F.Flandoli defined random attractor for random dynamical systems by the definition of attractor. Thus, the theory of attractor got very good development.In this paper, We'll consider the stochastic two-dimensional Boussinesq equation perturbed by an additive white noise. The deterministic case of Boussinesq equation has been studied systematically by many authors (see [1-5]). It is necessary to add to the equation a random force-additive white noise, we can study the stachastic case. The Boussinesq equation is mathematics model of thermohydraulics, which consists of fluid and temperature in the Bossinesq approximation. In the second chapter of this paper, we prove the existence and uniqueness of the global solution, and show that the solution continuously depended on the initial value. We also get some regularity results of the solutions. In the third chapter of this paper, the unique solution of equation (1-2) will generate a random dynamical system (RDS), this allows us to consider the random attractor of the RDS. To study stochastic equations, ones have to take someadaptive change of variable. A simple change of variable v=ζ+ω(hereω=sum from j=1 to mΦ_j(x)W_j(t)does not woke for the equations. We'll introduce a new variable which involves Ornstein-Uhlenbeck processes, and chang the stachastic case to the deterministic case.we also need some Sobolev norm estimates on the operators in the deterministic case. We can get:Lemma 2.3.1 The bilinear operator B:V×V→V′and D(A)×D(A)→H is continuous and satisfies:where c₁,c₂, c₃ are appropriate constants andθ_i∈[0,1), i=1,2,3. Lemma 2.3.2 The operator R(t): V→V′and D(A)→H satisfies |R(t)u|≤C₂(t)‖u‖+C₃(t),(?) u∈V (?) t≥0,where C₁(t)=β|sum from j=0 to m O_j(t)|,C₂(t)=4 2^1/2,C₃(t)=4C₁²(t)+(α+3)C₁(t),C₁(t),C₁(t),C₂(t),and C₃(t) are continuous in t and at most polynomial growth at t→∞, and|(R(t)u,u)|≤C₂(t)‖u(?)u|+C₃(t)|u|,(?) u∈V,(?) t≥0．Lemma 2.3.3 The bilinear form a on V×V satisfieswhere C4, C5 are appropriate constants.From above, we can get the existence and uniqueness of the global solution, and show that the solution continuously depended on the initial value. We also get some regularity results of the solutions.Theorem 2.4 For P-a.s.ω∈Ωand t₀∈R, there exists an unique solution denoted by u(t) of the equation (15) with u(t₀)=u₀, such that(i) if u₀∈H, thenu∈C([t₀,∞),H)∩L_loc²(t₀,∞；V)ï¼›and the mapping u₀→u(t) is continuous form H into D(A), for every t>t₀.(ii) if u₀∈V, thenu∈C([t₀,∞),V)∩L_loc²(t₀,∞；D(A))．From Theorem 2.4, we can get the random dynamical system and prove the following in the third part:Lemma 3.2.1 Let u={ξ,η} be a solution of (11), then|η(t)|≤(|η(t₀)|+c₁^′)e^{2(t₀-t)+c₁′,(?)t≥t₀．where c₁′=2T₀|D|1/2 is a determine constant.Lemma 3.2.2 exist a number C and a random variableγ(ω) satisfying the follow property: For everyρ>0, there exists t(ω,p)≤-1, such that for any b₀={v₀,η₀}∈H with |b₀|≤ρ, and for any t₀≤t(ω,ρ),the following estimates hold P-a.s. In particular,B(0,C+γ(ω)+|z(0)|) is a random absorbing set for RDS (?) in H.Lemma3.2.3 the same assumption and notations of Lemma 3.2.2, there exist two random variablesγ₁(ω) andγ₂(ω) such that for t₀≤t(ω,p),Lemma 3.2.4 There exist two random random radiusγ₃(ω),γ₄(ω) satisfying the following properties: For allρ>0 there exist t(ω,ρ)≤-1 such that the following holds P-a.s.,for all t₀≤t(ω,ρ) and b₀={v₀,η₀)∈H with |b₀|<ρ,From the Lemma(3.2.1-3.2.4)and the Theorem 1.3.18 ,we can obtain final conclusion:Theorem3.2.5 The random dynamical system associated with the stochastic Boussinesqequations (1-2)has a compact global attractor, which is contained in a random ball of V andattracts all deterministic bounded sets of H.