The asymptotic behavior of primitive equations with multiplicative noise
aa r X i v : . [ m a t h . P R ] J a n The asymptotic behavior of primitiveequations with multiplicative noise ∗ Rangrang Zhang † Guoli Zhou ‡ October 10, 2018
Abstract
This paper is concerned with the existence of invariant measure for 3D stochasticprimitive equations driven by linear multiplicative noise under non-periodic boundaryconditions. The common method is to apply Sobolev imbedding theorem to provingthe tightness of the distribution of the solution. However, this method fails becauseof the non-linearity and non-periodic boundary conditions of the stochastic primitiveequations. To overcome the difficulties, we show the existence of random attractor byproving the compact property and the regularity of the solution operator. Then weshow the existence of invariant measure.
Keywords:
Stochastic primitive equations, Random attractor, Invariant measure
Mathematics Subject Classification (2000):
The paper is concerned with the stochastic primitive equations (PEs) in a cylindrical domain O = M × ( − h, ⊂ R , where M is a smooth bounded domain in R : ∂ t υ + ( υ · ∇ ) υ + w∂ z υ + f υ ⊥ + ∇ p + L υ = n X k =1 α k υ ◦ dw k , (1.1) ∂ z p + T = 0 , (1.2) ∇ · υ + ∂ z w = 0 , (1.3) ∂ t T + υ · ∇ T + w∂ z T + L T = Q + n X k =1 β k T ◦ dw k . (1.4) ∗ This work was partially supported by NNSF of China(Grant No. 11401057), Natural Science Founda-tion Project of CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central Universi-ties(Grant No. 106112015CDJXY100005) and China Scholarship Council (Grant No.:201506055003). † Beijing Institute of Technology, No. 5 Zhongguancun South Street, Haidian District, Beijing, 100081, PR [email protected]. ‡ Chongqing University, No.174 Shazheng Street, Shapingba District, Chongqing, 400044, P R ChinaE-mail:[email protected]. he unknowns for the 3D stochastic PEs are the fluid velocity field ( υ, w ) = ( υ , υ , w ) ∈ R with υ = ( υ , υ ) and υ ⊥ = ( − υ , υ ) being horizontal, the temperature T and the pressure p. f = f ( β + y ) is the given Coriolis parameter, Q is a given heat source. ∇ = ( ∂ x , ∂ y ) is the horizontal gradientoperator and ∆ = ∂ x + ∂ y is the horizontal Laplacian. Let α i , β i ∈ R , i = 1 , , · · · , n , { w i , w i , i =1 , ...n } be a sequence of one-dimensional, independent, identically distributed Brownian motions.Here, we take P nk =1 α k υ ◦ dw k and P nk =1 β k T ◦ dw k to be Stratonovich multiplicative noise.The viscosity and the heat diffusion operators L and L are given by L i = − ν i ∆ − µ i ∂ zz , i = 1 , , where ν , µ are the horizontal and vertical Reynolds numbers and ν , µ are the horizontal andvertical heat diffusivity. Without loss of generality, we assume that ν = µ = ν = µ = 1 . The boundary of O is partitioned into three parts: Γ u ∪ Γ b ∪ Γ s , whereΓ u = { ( x, y, z ) ∈ O : z = 0 } , Γ b = { ( x, y, z ) ∈ O : z = − h } , Γ s = { ( x, y, z ) ∈ O : ( x, y ) ∈ ∂M, − h ≤ z ≤ } . Here h is a sufficiently smooth function. Without loss of generality, we assume h = 1 . We impose the following boundary conditions on the 3D stochastic PEs. ∂ z υ = η, w = 0 , ∂ z T = − γ ( T − τ ) on Γ u , (1.5) ∂ z υ = 0 , w = 0 , ∂ z T = 0 on Γ b , (1.6) υ · ~n = 0 , ∂ ~n υ × ~n = 0 , ∂ ~n T = 0 on Γ s , (1.7)where η ( x, y ) is the wind stress on the surface of the ocean, γ is a positive constant, τ is the typicaltemperature distribution on the top surface of the ocean and ~n is the norm vector to Γ s . For thesake of simplicity, we assume that Q is independent of time and η = τ = 0. It is worth pointingout that all results obtained in this paper are still valid for the general case by making some simplemodifications.Integrating (1.3) from − z and using (1.5), (1.6), we have w ( t, x, y, z ) := Φ( υ )( t, x, y, z ) = − Z z − ∇ · υ ( t, x, y, z ′ ) dz ′ , (1.8)moreover, Z − ∇ · υdz = 0 . Integrating (1.2) from − z , set p s be a certain unknown function at Γ b satisfying p ( x, y, z, t ) = p s ( x, y, t ) − Z z − T ( x, y, z ′ , t ) dz ′ . Then (1.1)-(1.4) can be rewritten as ∂ t υ + L υ + ( υ · ∇ ) υ + Φ( v ) ∂ z υ + ∇ p s − R z − ∇ T dz ′ + f υ ⊥ = P nk =1 α k υ ◦ dw k , (1.9) ∂ t T + L T + υ · ∇ T + Φ( v ) ∂ z T = Q + P nk =1 β k T ◦ dw k , (1.10) R − ∇ · υdz = 0 . (1.11) he boundary value and initial conditions for (1.9)-(1.11) are given by ∂ z υ | Γ u = ∂ z υ | Γ b = 0 , υ · ~n | Γ s = 0 , ∂ ~n υ × ~n | Γ s = 0 , (1.12) (cid:16) ∂ z T + γT (cid:17) | Γ u = ∂ z T | Γ b = 0 , ∂ ~n T | Γ s = 0; (1.13) υ ( x, y, z, t ) = υ ( x, y, z ) , T ( x, y, z, t ) = T ( x, y, z ) . (1.14)The primitive equations are the basic model used in the study of climate and weather predic-tion, which can be used to describe the motion of the atmosphere when the hydrostatic assumptionis enforced (see [15, 19, 20] and the references therein). This model has been intensively investi-gated because of the interests stemmed from physics and mathematics. As far as we know, theirmathematical study was initiated by Lions, Temam and Wang (see e.g. [25, 26, 27, 28]). Forexample, the existence of global weak solutions for the primitive equations was established in [26].Guill´en-Gonz´alez et al. obtained the global existence of strong solutions to the primitive equationswith small initial data in [18]. The local existence of strong solutions to the primitive equationsunder the small depth hypothesis was established by Hu et al. in [21]. Cao and Titi developeda beautiful approach to dealing with the L -norm of the fluctuation ˜ v of horizontal velocity andobtained the global well-posedness for the 3D viscous primitive equations in [9]. Subsequently,in [23], Kukavica and Ziane developed a different method to handle non-rectangular domains andboundary conditions with physical reality. The existence of the global attractor was establishedby Ju in [22]. For the global well-posedness of 3D primitive equations with partial dissipation, werefer the reader to some papers, see e.g. [5 , , , , H ( O ). However, it is quite difficult because of highly non-linear drift terms andnon-periodic boundary conditions. Instead of using that method, we provide a new method to finda compact absorbing set in H ( O ), which guarantees the existence of random attractor in H ( O ).The main idea is that we firstly prove the solution operator of 3D stochastic PEs is compact in H ( O ), P − a.s.. Then we can construct a compact absorbing ball in H ( O ) by the compact propertyof solution operator. It’s important to point out that the random attractor we obtained is strongerthan that in [16]. Specifically, the random attractor we obtain is compact in H ( O ), P -a.s. andattracts any orbit starting from −∞ in the strong topology of H ( O ) while the random attractorin [16] is not necessarily a compact subset in H ( O ). In addition, the linear multiplicative noiseas well as non-periodic boundary conditions are handled at the same time in this paper, which aredifferent from [17].Taking into account the asymptotical compact property of the solution operator, we can provethe existence of invariant measures by showing that the one-point motions associated with the flowgenerated by 3D PEs define a family of Markov processes. Up to now, there are no works on the xistence of invariant measures for the stochastic PEs subject to non-periodic boundary conditions.Maybe it is an attempt to solve this problem by proving the asymptotic compact property of thesolution operator.The remaining of this paper is organized as follows. In section 2, some preliminaries of 3Dstochastic primitive equations are stated. In section 3, the global well-posedness of 3D stochasticprimitive equations is proved. In section 4, we establish the existence of random attractor. Finally,in section 5, the existence of invariant measures for 3D stochastic primitive equations is obtained.As usual, constants C may change from one line to the next, unless, we give a special declaration.Denote by C ( a ) a constant which depends on some parameter a . For 1 ≤ p ≤ ∞ , let L p ( O ) , L p ( M ) be the usual Lebesgue spaces with the norm | φ | p = ( (cid:0)R O | φ ( x, y, z ) | p dxdydz (cid:1) p , φ ∈ L p ( O ) , (cid:0)R M | φ ( x, y ) | p dxdy (cid:1) p , φ ∈ L p ( M ) . In particular, | · | and ( · , · ) represent norm and inner product of L ( O ) (or L ( M )), respectively.For m ∈ N + , ( W m,p ( O ) , k · k m,p ) stands for the classical Sobolev space, see [1]. When p = 2, wedenote by H m ( O ) = W m, ( O ) with norm k · k m . Without confusion, we shall sometimes abusenotation and denote by k · k m the norm in H m ( M ). Let V = n υ ∈ ( C ∞ ( O )) : ∂ z υ | z =0 = 0 , ∂ z υ | z = − = 0 , υ · ~n | Γ s = 0 ,∂ ~n υ × ~n | Γ s = 0 , Z − ∇ · υdz = 0 o , V = n T ∈ C ∞ ( O ) : ∂ z T | z = − = 0 , ( ∂ z T + γT ) | z =0 = 0 , ∂ ~n T | Γ s = 0 o . We denote by V the closure space of V in ( H ( O )) and V the closure space of V in H ( O ),respectively. Let H be the closure space of V with respect to the norm | · | . Define H = L ( O ) . Set V = V × V , H = H × H . Let U = ( υ, T ), ˜ U = (˜ υ, ˜ T ), V is equipped with the inner product h U, ˜ U i V := h υ, ˜ υ i V + h T, ˜ T i V , h υ, ˜ υ i V := Z O (cid:18) ∇ v · ∇ ˜ v + ∂v∂z · ∂ ˜ v∂z (cid:19) dxdydz, h T, ˜ T i V := Z O ∇ T · ∇ ˜ T + ∂T∂z ∂ ˜ T∂z ! dxdydz + γ Z Γ u T T dxdy. Subsequently, the norm of V is defined by k U k = h U, U i V . The inner product of H is defined by h U, ˜ U i H := h υ, ˜ υ i + h T, ˜ T i , h υ, ˜ υ i := Z O υ · ˜ υdxdydz, h T, ˜ T i := Z O T ˜ T dxdydz.
Denote V ′ i the dual space of V i , i = 1 , . Furthermore, we have the compact embedding relationship D ( A i ) ⊂ V i ⊂ H i ⊂ V ′ i ⊂ D ( A i ) ′ , nd h· , ·i V i = h A i · , ·i = h A i · , A i ·i , i = 1 , . For the sake of simplicity, in the following, we denote Z O · dxdydz = Z O · , Z M · dxdy = Z M · . In this section, we aim to prove the global well-posedness of (1.9)-(1.14). Firstly, we introduce thefollowing definition. Given T > , fix a single stochastic basis (Ω , F , {F t ,t } t ∈ [ t , T ] , P ), where F t ,t := σ ( W jk ( s ) − W jk ( t ) , s ∈ [ t , t ] , j = 1 , . (3.15) Definition 3.1
Fix T > , a continuous V − valued F t ,t − adapted random field ( U ( ., t )) t ∈ [ t , T ] =( υ ( ., t ) , T ( ., t )) t ∈ [ t , T ] defined on (Ω , F , P ) is said to be a strong (weak) solution to (1.9)-(1.14) if U ∈ C ([ t , T ]; V ) ∩ L ([ t , T ]; ( H ( O )) ) ( U ∈ C ([ t , T ]; H ) ∩ L ([ t , T ]; ( H ( O )) ) P − a.s.. and the following Z O υ ( t ) · φ − Z tt ds Z O { [( υ · ∇ ) φ + Φ( υ ) ∂ z φ ] υ − [( f k × υ ) · φ + ( Z z − T dz ′ ) ∇ · φ ] } + Z tt ds Z O υ · L φ = Z O υ · φ + Z tt Z O n X k =1 α k υ ◦ dw k ( s, w ) · φ , Z O T ( t ) φ − Z tt ds Z O [( υ · ∇ ) φ + Φ( υ ) ∂ z φ ] T + Z tt ds Z O T L φ = Z O T φ + Z tt ds Z O Qφ + Z tt Z O n X k =1 β k T ◦ dw k ( s, w ) · φ , holds P − a.s., for all t ∈ [ t , T ] and φ = ( φ , φ ) ∈ D ( A ) × D ( A ) . Consider α ( t ) = exp( − n X k =1 α k w k ) , β ( t ) = exp( − n X k =1 β k w k ) . Then α ( t ) and β ( t ) satisfy the following stratonovich equations dα ( t ) = − n X k =1 α k α ( t ) ◦ dw k ( t ) , dβ ( t ) = − n X k =1 β k β ( t ) ◦ dw k ( t ) . Define ( u ( t ) , θ ( t )) = ( α ( t ) v ( t ) , β ( t ) T ( t )) . Then, ( u ( t ) , θ ( t )) satisfies ∂ t u − ∆ u − ∂ zz u + α − u · ∇ u + α − Φ( u ) ∂ z u + f u ⊥ + α ∇ p s − αβ − R z − ∇ θdz ′ = 0 , (3.16) ∂ t θ − ∆ θ − ∂ zz θ + α − u · ∇ θ + α − Φ( u ) ∂ z θ = βQ, (3.17) R − ∇ · udz = 0 . (3.18) he boundary and initial conditions for (3.16)-(3.18) are ∂ z u | Γ u = ∂ z u | Γ b = 0 , u · ~n | Γ s = 0 , ∂ ~n u × ~n | Γ s = 0 , (3.19) (cid:16) ∂ z θ + γθ (cid:17) | Γ u = ∂ z θ | Γ b = 0 , ∂ ~n θ | Γ s = 0 , (3.20)( u (cid:12)(cid:12) t , θ (cid:12)(cid:12) t ) = ( υ , T ) . (3.21) Definition 3.2
Let T be a fixed positive time and ( v , T ) ∈ V . ( u, θ ) is called a strong solution ofthe system (3.16)-(3.21) on the time interval [ t , T ] if it satisfies (3.16)-(3.21) in the weak sensesuch that P -a.s. u ∈ C ([ t , T ]; V ) ∩ L ([ t , T ]; ( H ( O )) ) ,θ ∈ C ([ t , T ]; V ) ∩ L ([ t , T ]; H ( O )) . Theorem 3.1 (Existence of local solutions to (3.16)-(3.21)) If Q ∈ L ( O ) , v ∈ V , T ∈ V . Then, for P -a.s., ω ∈ Ω , there exists a stopping time T ∗ > such that ( u, θ ) is a strongsolution of the system (3.16)-(3.21) on the interval [ t , T ∗ ] . The proof of the existence of local solutions to (3.16)-(3.21) is similar to [18], we omit it. Beforeshowing the global well-posedness of the strong solution, we recall the following Lemma, a specialcase of a general result of Lions and Magenes [24], which will help us to show the continuity of thesolution with respect to time in ( H ( O )) . For the proof, see [29] . Lemma 3.1
Let
V, H, V ′ be three Hilbert spaces such that V ⊂ H = H ′ ⊂ V ′ , where H ′ and V ′ are the dual spaces of H and V , respectively. Suppose u ∈ L (0 , T ; V ) and dudt ∈ L (0 , T ; V ′ ) . Then u is almost everywhere equal to a continuous function from [0 , T ] into H . Theorem 3.2 [ Existence of global solution to (3.16)-(3.21) ] If Q ∈ L ( O ) , v ∈ V , T ∈ V ,and T > . Then, for P -a.s., ω ∈ Ω , there exists a unique strong solution ( u, θ ) of the system(3.16)-(3.21) or equivalently ( v, T ) to the system (1.9)-(1.14) on the interval [ t , T ] . Proof.
Let [ t , τ ∗ ) be the maximal interval of existence of the strong solution, for fixed ω ∈ Ω , we will establish various norms of this solution in the interval [ t , τ ∗ ) . In particular, we will showthat if τ ∗ < ∞ then H − norm of the strong solution is bounded over the interval [ t , τ ∗ ). A priori estimates:
Referring to [9], define¯ φ ( x, y ) = Z − φ ( x, y, ξ ) dξ, ∀ ( x, y ) ∈ M. In particular, ¯ u ( x, y ) = Z − u ( x, y, ξ ) dξ, in M. Let ˜ u = u − ¯ u. Notice that ¯˜ u = 0 , ∇ · ¯ u = 0 in M. aking the average of equations (3.16) in the z direction over the interval ( − , ∂ t ¯ u + α − u · ∇ u + Φ( u ) ∂ z u + f ¯ u ⊥ + α ∇ p s − αβ − Z − Z z − ∇ θdz ′ dz − ∆¯ u = 0 . (3.22)By the integration by parts, we get Z − Φ( u ) ∂ z udz = Z − u ∇ · udz = Z − ( ∇ · ˜ u )˜ udz, (3.23) Z − u · ∇ udz = Z − ˜ u · ∇ ˜ udz + ¯ u · ∇ ¯ u. (3.24)Substituting (3.23) and (3.24) into (3.22), ¯ u satisfies ∂ t ¯ u − ∆¯ u + α − (˜ u · ∇ ˜ u + ˜ u ∇ · ˜ u + ¯ u · ∇ ¯ u ) + f ¯ u ⊥ + α ∇ p s − αβ − ∇ R − R z − θ ( x, y, λ, t ) dλdz = 0 , (3.25) ∇ · ¯ u = 0 in M, (3.26)¯ u · ~n = 0 , ∂ ~n ¯ u × ~n = 0 on M. (3.27)By subtracting (3.25) from (3.16) and using (3.19), (3.27), we conclude that ˜ u satisfies ∂ t ˜ u − ∆˜ u − ∂ zz ˜ u + α − ˜ u · ∇ ˜ u + α − Φ(˜ u ) ∂ z ˜ u + α − ˜ u · ∇ ¯ u + α − ¯ u · ∇ ˜ u − α − ˜ u · ∇ ˜ u − α − ˜ u ∇ · ˜ u + f ˜ u ⊥ − αβ − R z − ∇ θdz ′ + αβ − R − R z − ∇ θdz ′ dz = 0 , (3.28) ∂ z ˜ u | z =0 = 0 , ∂ z ˜ u | z = − = 0 , ˜ u · ~n | Γ s = 0 , ∂ ~n ˜ u × ~n | Γ s = 0 . (3.29)In the following, we will study the properties of ¯ u and ˜ u. (1) Estimates of | θ | and | u | . Take the inner product of equation (3.17) with θ in H , weget 12 ∂ t | θ | + |∇ θ | + | θ z | + γ | θ ( z = 0) | = β Z O Qθ − α − Z O (cid:16) u · ∇ θ + Φ( u ) ∂ z θ (cid:17) θ. By integration by parts, α − Z O (cid:16) u · ∇ θ + Φ( u ) ∂ z θ (cid:17) θ = 0 . By the H¨older inequality, we have12 ∂ t | θ | + |∇ θ | + | θ z | + γ | θ ( z = 0) | ≤ β Z O Qθ ≤ ε | θ | + Cβ | Q | , referring to (48) in [9], | θ | ≤ | ∂ z θ | + 2 | θ ( z = 0) | , then, we arrive at ∂ t | θ | + 2 |∇ θ | + (2 − ε ) | θ z | + (2 γ − ε ) | θ ( z = 0) | ≤ Cβ | Q | . (3.30)Hence, there exists a positive λ such that ∂ t | θ | + λ | θ | ≤ Cβ | Q | . pplying the Gronwall inequality, we have | θ ( t ) | ≤ | θ t | e − λ ( t − t ) + C Z tt β e λ ( s − t ) | Q | ds. (3.31)In view of (3.30) and (3.31), we obtainsup t ∈ [ t ,τ ∗ ) | θ ( t ) | + Z τ ∗ t k θ ( t ) k dt ≤ C. (3.32)Taking inner product of (3.16) with u in H , by integration by parts, we have ∂ t | u | + (1 − ε ) |∇ u | + | u z | ≤ C | θ | . (3.33)By (3.32), we get sup t ∈ [ t ,τ ∗ ) | u ( t ) | + Z τ ∗ t k u ( t ) k dt ≤ C. (3.34) (2) Estimates of | θ | and | ˜ u | . Taking the inner product of the equation (3.17) with θ in H , we have 14 ∂ t | θ | + 34 |∇ θ | + 34 | ( θ ) z | + γ Z M | θ ( z = 0) | = β Z O Qθ − α − Z O [ u · ∇ θ + Φ( u ) ∂ z θ ] θ . (3.35)By integration by parts, we have α − Z O [ u · ∇ θ + Φ( u ) ∂ z θ ] θ = 0 . (3.36)Applying the interpolation inequality to | θ | , we obtain | θ | ≤ C | θ | ( |∇ θ | + | ∂ z θ | + α | θ ( z = 0) | ) . (3.37)Using (3.37) and the H¨older inequality, we get Z O βQθ ≤ ε ( |∇ θ | + | ∂ z θ | + α | θ ( z = 0) | ) + Cβ | Q | | θ | . (3.38)Combining (3.36) and (3.38), we arrive at ∂ t | θ | + |∇ θ | + | ( θ ) z | + α Z M | θ ( z = 0) | ≤ Cβ | Q | | θ | . (3.39)Since θ ( x, y, z ) = − Z z ∂ r θ ( x, y, r ) dr + θ ( z = 0) , by the Young’s inequality, we have | θ | = − Z O Z z ∂ r θ dr + Z M Z − θ ( z = 0) dz ≤ | θ | + 8 | ∂ z ( θ ) | + Z M θ ( z = 0) , hen | θ | ≤ | ∂ z ( θ ) | + 2 | θ ( z = 0) | . From (3.39), we get ∂ t | θ | + | θ | ≤ Cβ | Q | | θ | ,∂ t | θ | + | θ | ≤ Cβ | Q | | θ | . Applying the Gronwall inequality, there exists a positive number which is still denoted by λ suchthat | θ ( t ) | ≤ | θ t | e − λ ( t − t ) + C Z tt β ( s ) e − λ ( t − s ) | Q | ds (3.40)for t ∈ [ t , τ ∗ ) . Taking the inner product of the equation (3.28) with | ˜ u | ˜ u in H , we obtain14 ∂ t | ˜ u | + 12 Z O (cid:16) |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | (cid:17) + Z O | ˜ u | ( |∇ ˜ u | + | ∂ z ˜ u | )= − α − Z O ((˜ u · ∇ )˜ u + Φ(˜ u ) ∂ z ˜ u ) · | ˜ u | ˜ u − α − Z O (˜ u · ∇ ¯ u ) · | ˜ u | ˜ u − α − Z O (¯ u · ∇ ˜ u ) · | ˜ u | ˜ u + α − Z O ˜ u ∇ · ˜ u + ˜ u · ∇ ˜ u · | ˜ u | ˜ u + αβ − Z O ( Z z − ∇ θdz ′ − Z − Z z − ∇ θdz ′ dz ) · | ˜ u | ˜ u. By integration by parts and boundary conditions (3.29), we have Z O ((˜ u · ∇ )˜ u + Φ(˜ u ) ∂ z ˜ u ) | ˜ u | ˜ u = 0 , Z O (¯ u · ∇ ˜ u ) | ˜ u | ˜ u = − Z O | ˜ u | ∇ · ¯ u = 0 , and Z O [(˜ u · ∇ )¯ u ] · | ˜ u | ˜ u = − Z O [(˜ u · ∇ ) | ˜ u | ˜ u ] · ¯ u − Z O ( ∇ · ˜ u ) | ˜ u | ˜ u · ¯ u, Z O ˜ u ∇ · ˜ u + ˜ u · ∇ ˜ u · | ˜ u | ˜ u = − Z O ˜ u k ˜ u j ∂ x k ( | ˜ u | ˜ u j ) , where ˜ u k is the k − th coordinate of ˜ u , k = 1 , . Based on the above equalities and by integration by parts, we obtain14 ∂ t | ˜ u | + 12 Z O (cid:16) |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | (cid:17) + Z O | ˜ u | ( |∇ ˜ u | + | ∂ z ˜ u | )= α − Z O ¯ u · (˜ u · ∇ ) | ˜ u | ˜ u + α − Z O ( ∇ · ˜ u )¯ u · | ˜ u | ˜ u − α − Z O ˜ u k ˜ u j ∂ x k ( | ˜ u | ˜ u j ) − αβ − Z O (cid:16) Z z − θdλ − Z − Z z − θdλdz (cid:17) ∇ · | ˜ u | ˜ u := I + I + I + I . (3.41) pplying the H¨older inequality, the Minkowski inequality and the interpolation inequalities, weobtain I ≤ α − Z M | ¯ u | Z − | ˜ u ||∇ ˜ u || ˜ u | dz ≤ α − Z M | ¯ u | (cid:16) Z − | ˜ u | |∇ ˜ u | dz (cid:17) (cid:16) Z − | ˜ u | dz (cid:17) ≤ α − | ¯ u | L ( M ) |∇ ( | ˜ u | ) | (cid:16) Z M ( Z − | ˜ u | dz ) (cid:17) ≤ α − | ¯ u | L ( M ) |∇ ( | ˜ u | ) | (cid:16) Z − ( Z M | ˜ u | ) dz (cid:17) ≤ α − | ¯ u | L ( M ) |∇ ( | ˜ u | ) | (cid:16) Z − | ( | ˜ u | ) | L ( M ) k ( | ˜ u | ) k H ( M ) dz (cid:17) ≤ α − | ¯ u | L ( M ) |∇ ( | ˜ u | ) || ( | ˜ u | ) | [ | ( | ˜ u | ) | + |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | ] . By the Young’s inequality and the interpolation inequalities, we get I ≤ ε ( |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | ) + C ( α − | ¯ u | L ( M ) + α − | ¯ u | L ( M ) ) | ˜ u | ≤ ε ( |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | ) + C ( α − k u k + α − | u | k u k ) | ˜ u | . (3.42)By the H¨older inequality, the Minkowski inequality, the interpolation inequality and the Young’sinequality, we get I = α − Z M ¯ u · Z − |∇ ˜ u || ˜ u | dz ≤ α − Z M | ¯ u | (cid:16) Z − |∇ ˜ u | | ˜ u | dz (cid:17) (cid:16) Z − | ˜ u | dz (cid:17) ≤ α − ||∇ ˜ u || ˜ u ||| ¯ u | (cid:16) Z M (cid:16) Z − | ˜ u | dz (cid:17) (cid:17) ≤ α − ||∇ ˜ u || ˜ u ||| ¯ u | (cid:16) Z − (cid:16) Z M | ˜ u | × (cid:17) dz (cid:17) ≤ Cα − ||∇ ˜ u || ˜ u ||| ¯ u | (cid:16) Z − || ˜ u | | L ( M ) ( |∇| ˜ u | | L ( M ) + | ∂ z | ˜ u | | L ( M ) + || ˜ u | | L ( M ) ) dz (cid:17) ≤ ε ( ||∇ ˜ u || ˜ u || + |∇| ˜ u | | + | ∂ z | ˜ u | | ) + C ( α − | ¯ u | + α − | ¯ u | ) || ˜ u | | ≤ ε ( ||∇ ˜ u || ˜ u || + |∇| ˜ u | | + | ∂ z | ˜ u | | ) + C ( α − k u k + α − | u | k u k ) | ˜ u | . Applying the H¨older inequality, the Minkowski inequality, the interpolation inequality and the oung’s inequality, we have I = − α − Z O ˜ u k ˜ u j ∂ x k ( | ˜ u | ˜ u j ) ≤ α − Z M (cid:16) Z − | ˜ u | dz (cid:17)(cid:16) Z − |∇ ˜ u || ˜ u | dz (cid:17) ≤ α − (cid:16) Z O |∇ ˜ u | | ˜ u | (cid:17) (cid:16) Z M (cid:16) Z − | ˜ u | dz (cid:17) (cid:17) ≤ α − (cid:16) Z O |∇ ˜ u | | ˜ u | (cid:17) (cid:16) Z − (cid:16) Z M | ˜ u | (cid:17) dz (cid:17) ≤ Cα − (cid:16) Z O |∇ ˜ u | | ˜ u | (cid:17) (cid:16) Z − | ˜ u | L ( M ) · k ˜ u k H ( M ) dz (cid:17) ≤ Cα − (cid:16) Z O |∇ ˜ u | | ˜ u | (cid:17) (cid:16) Z − | ˜ u | L ( M ) dz (cid:17) Z − k ˜ u k H ( M ) dz ≤ ε Z O |∇ ˜ u | | ˜ u | + Cα − k u k | ˜ u | . Analogously, we have I ≤ αβ − (cid:16) Z O |∇ ˜ u | | ˜ u | (cid:17) (cid:16) Z O | ˜ u | (cid:17) (cid:16) Z O | θ | (cid:17) ≤ ε Z O |∇ ˜ u | | ˜ u | + Cα β − | ˜ u | | θ | . Collecting all the above inequalities, we have ∂ t | ˜ u | + Z O (cid:16) |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | (cid:17) + Z O | ˜ u | ( |∇ ˜ u | + | ∂ z ˜ u | ) ≤ Cα β − | θ | | ˜ u | + C ( α − k u k + α − | u | k u k ) | ˜ u | , (3.43)and ∂ t | ˜ u | ≤ Cα β − | θ | + C ( α − k u k + α − | u | k u k ) | ˜ u | . (3.44)Applying the Gronwall inequality, we conclude thatsup t ∈ [ t ,τ ∗ ) | ˜ u ( t ) | + Z τ ∗ t Z O (cid:16) |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | (cid:17) ds + Z τ ∗ t Z O | ˜ u | ( |∇ ˜ u | + | ∂ z ˜ u | ) ds ≤ C. (3.45) (3) Estimates of |∇ ¯ u | and | u z | . Taking the inner product of equation (3.25) with − ∆¯ u in L ( M ), we arrive at12 ∂ t |∇ ¯ u | + | ∆¯ u | = − α − Z M ¯ u · ∇ ¯ u · ∆¯ u − α − Z M ˜ u · ∇ ˜ u · ∆¯ u − α − Z M ˜ u ∇ · ˜ u · ∆¯ u − Z M (¯ u ⊥ + α ∇ p s ) · ∆¯ u + αβ − Z M ∇ Z − Z z − θ ( x, y, λ, t ) dλdz · ∆¯ u. y integration by parts and (3.26)-(3.27)(for more detail, see [9] ), we have Z M ¯ u ⊥ · ∆¯ u = 0 , Z M ∇ p s · ∆¯ u = 0 , Z M ∇ Z − Z z − θ ( x, y, λ, t ) dλdz · ∆¯ u = 0 . Applying the H¨older inequality and the interpolation inequalities, we obtain α − Z M ¯ u · ∇ ¯ u · ∆¯ u ≤ Cα − | ¯ u | |∇ ¯ u || ∆¯ u | ≤ ε | ∆¯ u | + Cα − | ¯ u | |∇ ¯ u | . Using the H¨older inequality, the Minkowski inequality and the Sobolev imbedding theorem, wehave α − Z M ˜ u ∇ · ˜ u · ∆¯ u + α − Z M ˜ u · ∇ ˜ u · ∆¯ u ≤ α − | ∆¯ u | (cid:16) Z O | ˜ u | |∇ ˜ u | (cid:17) ≤ ε | ∆¯ u | + Cα − Z O | ˜ u | |∇ ˜ u | . From the above inequalities, we conclude that ∂ t |∇ ¯ u | + | ∆¯ u | ≤ Cα − | u | k u k |∇ ¯ u | + Cα − Z O | ˜ u | |∇ ˜ u | . (3.46)Therefore, applying Gronwall inequality and (3.34), (3.45), (3.46), we arrive atsup t ∈ [ t ,τ ∗ ) |∇ ¯ u ( t ) | ≤ C. (3.47)Denote u z = ∂u∂z , from (3.16), we get ∂ t u z − ∆ u z − ∂ zz u z + α − u · ∇ u z + α − u z · ∇ u − α − u z ∇ · u + α − Φ( u ) u zz + f k × u z − αβ − ∇ θ = 0 . (3.48)Taking the inner product of the equation (3.48) with u z in H , we obtain12 ∂ t | u z | + |∇ u z | + | u zz | = − α − Z O (cid:16) ( u · ∇ ) u z + Φ( u ) u zz (cid:17) · u z − α − Z O [( u z · ∇ ) u ] · u z + α − Z O ( ∇ · u ) u z · u z − Z O u ⊥ z · u z + αβ − Z O ∇ θ · u z . By integration by parts, we obtain α − Z O (cid:16) ( u · ∇ ) u z + Φ( u ) u zz (cid:17) · u z = 0 . Thanks to the H¨older inequality, the interpolation inequality and the Sobolev imbedding theorem,we reach α − Z O [( u z · ∇ ) u ] · u z ≤ Cα − Z O | u || u z ||∇ u z |≤ Cα − |∇ u z || u | | u z | ≤ Cα − |∇ u z || u | | u z | ( |∇ u z | + | ∂ z u z | + | u z | ) ≤ ε ( |∇ u z | + | ∂ z u z | ) + C ( α − |∇ ¯ u | + 1) | u z | . imilar to the above, we get Z O ( ∇ · u ) u z · u z ≤ ε ( |∇ u z | + | u zz | ) + C ( α − |∇ ¯ u | + 1) | u z | . Collecting the above inequalities, we have ∂ t | u z | + |∇ u z | + | u zz | ≤ C ( α − | ¯ u | + α − |∇ ¯ u | + 1) | u z | . (3.49)Applying Gronwall inequality to (3.49), and by (3.47), we reachsup t ∈ [ t ,τ ∗ ) | u z ( t ) | + Z τ ∗ t ( |∇ u z ( s ) | + | u zz ( s ) | ) ds ≤ C. (3.50) (4) Estimates of |∇ u | and |∇ θ | . Taking the inner product of equation (3.16) with − ∆ u in H , we reach 12 ∂ t |∇ u | + | ∆ u | + |∇ ∂ z u | = α − Z O [( u · ∇ ) u + Φ( u ) ∂ z u ] · ∆ u + Z O f k × u · ∆ u + α Z O ∇ p s · ∆ u − αβ − Z O ( Z z − ∇ θdz ′ ) · ∆ u. By the H¨older inequality, the interpolation inequality and the Sobolev inequality, we have α − Z O [( u · ∇ ) u ] · ∆ u ≤ Cα − | ∆ u ||∇ u | | u | ≤ α − | ∆ u ||∇ u | ( | ∆ u | + |∇ u z | + |∇ u | ) | u | ≤ ε ( | ∆ u | + |∇ u z | ) + C ( α − | u | + α − | u | ) |∇ u | ≤ ε ( | ∆ u | + |∇ u z | ) + C ( α − | ˜ u | + α − |∇ ¯ u | + α − | ˜ u | + α − |∇ ¯ u | ) |∇ u | . (3.51)Due to the H¨older inequality, the Minkowsky inequality, the interpolation inequality and theSobolev embedding theorem, we get α − Z O Φ( u ) u z · ∆ u ≤ α − Z M (cid:16) Z − |∇ · u | dz Z − | u z | · | ∆ u | dz (cid:17) ≤ α − | ∆ u | (cid:16) Z M ( Z − |∇ · u | dz ) (cid:17) (cid:16) Z M ( Z − | u z | dz ) (cid:17) ≤ Cα − | ∆ u | (cid:16) Z − |∇ u | L ( M ) ( | ∆ u | L ( M ) + |∇ u | L ( M ) ) dz (cid:17) × (cid:16) Z − | u z | L ( M ) ( |∇ u z | L ( M ) + | u z | L ( M ) ) dz (cid:17) ≤ ε ( | ∆ u | + |∇ u z | ) + C |∇ u | ( α − | u z | + α − | u z ||∇ u z | + α − | u z | + α − | u z | |∇ u z | ) . (3.52)We also have Z O ( f k × u ) · ∆ u = 0 , Z O ∇ p s · ∆ u = 0 . ollecting all the above inequalities, we get ∂ t |∇ u | + | ∆ u | + |∇ ∂ z u | ≤ C ( α β − k θ k + α − | ˜ u | + α − | ˜ u | + α − |∇ ¯ u | + α − |∇ ¯ u | + α − | u z | + α − | u z | + α − |∇ u z | + α − | u z | |∇ u z | ) |∇ u | . (3.53)Applying the Gronwall inequality, and by (3.32),(3.45), (3.47), (3.50), we obtainsup t ∈ [ t ,τ ∗ ) |∇ u ( t ) | + Z τ ∗ t ( | ∆ u ( t ) | + |∇ ∂ z u | ) dt ≤ C. (3.54)Taking the inner product of the equation (3.17) with − ∆ θ − θ zz in H , similar to the above, we get12 ∂ t ( |∇ θ | + | θ z | + γ |∇ θ ( z = 0) | ) + | ∆ θ | + 2( |∇ θ z | + γ |∇ θ ( z = 0) | ) + | θ zz | = Z O ( α − u · ∇ θ + α − Φ( u ) θ z − βQ )(∆ θ + θ zz ) ≤ ε ( | ∆ θ | + | θ zz | + |∇ θ z | ) + Cβ | Q | + Cα − |∇ u | | ∆ u | | θ z | + C ( α − | ˜ u | + α − |∇ ¯ u | + α − | ˜ u | + α − |∇ ¯ u | ) |∇ θ | . Utilizing the Gronwall inequality, we get |∇ θ | + | θ z | + γ |∇ θ ( z = 0) | + Z tt [ | ∆ θ | + 2( |∇ θ z | + γ |∇ θ ( z = 0) | ) + | θ zz | ] ds ≤ C. (3.55)In the following, we will complete our proof of the global well-posedness of stochastic PEs bythree steps. Concretely, we firstly prove the global existence of strong solution. Then, we showthat the solution is continuous in the space V with respect to t . At last, we prove the continuityin V with respect to the initial data. Step 1: the global existence of strong solution.In the previous, we have obtained a priori estimates in V. As we have indicated before that [ t , τ ∗ )is the maximal interval of existence of the solution of (3.16)-(3.21), we infer that τ ∗ = ∞ , a.s..Otherwise, if there exists A ∈ F such that P ( A ) > ω ∈ A, τ ∗ ( ω ) < ∞ , it is clearthat lim sup t → τ −∗ ( ω ) ( k u ( t ) k + k θ ( t ) k ) = ∞ , for any ω ∈ A, which contradicts the priori estimates (3.50), (3.54) and (3.55). Therefore τ ∗ = ∞ , a.s., and thestrong solution ( u, θ ) exists globally in time a.s.. Step 2: the continuity of strong solutions with respect to t .Multiplying (3.16) by η ∈ V , integrating with respect to space variable, yields h ∂ t A u, η i = h ∂ t u, A η i = −h A u, A η i − α − h ( u · ∇ ) u, A η i− α − h Φ( u ) ∂ z u, A η i − h f u ⊥ , A η i + αβ − h Z z − ∇ θdz ′ , A η i , where h∇ p s , A η i = 0 is used. Taking a similar argument in (3.52), we get h Φ( u ) ∂ z u, A η i ≤ k u kk u k | A η | . y the H¨older inequality and the Sobolev imbedding theorem, we have k ∂ t ( A u ) k V ′ ≤ C ( k u k + k u kk u k + | u | + |∇ θ | ) . Since u ∈ L ∞ ([ t , T ]; V ) ∩ L ([ t , T ]; ( H ( O )) ) , ∀T > t , we obtain A u ∈ L ([ t , T ]; V ) , ∂ t ( A u ) ∈ L ([ t , T ]; V ′ ) . Referring to Lemma 3.1, we deduce that A u ∈ C ([ t , T ]; H ) or u ∈ C ([ t , T ]; V ) P − a.s.. To study the regularity of θ, we choose ξ ∈ V . By (3.17) we have h ∂ t A θ, ξ i = h ∂ t θ, A ξ i = h A θ, A ξ i − α − h u · ∇ θ, A ξ i + α − h Φ( u ) ∂ z θ, A ξ i + β h Q, A ξ i . Taking a similar argument as above, we get h Φ( u ) ∂ z θ, A ξ i ≤ C k u k k u k k θ k k θ k | A ξ | . Then by the H¨older inequality and the Sobolev imbedding theorem, we have | ∂ t A θ | V ′ ≤ C ( | A θ | + α − k u kk θ k + α − k u k k u k k θ k k θ k + β | Q | ) . In view of step one, we have θ ∈ L ∞ ([ t , T ]; V ) ∩ L ([ t , T ]; H ( O )) , ∀T > t . Therefore, by the same argument as above, we get A θ ∈ L ([ t , T ]; V ) , ∂ t ( A θ ) ∈ L ([ t , T ]; V ′ ) . We deduce from Lemma 3.1 that A θ ∈ C ([ t , T ]; H ) or θ ∈ C ([ t , T ]; V ) , P − a.s.. Step 3: the continuity in V with respect to the initial data.Let ( υ , T ) and ( υ , T ) be two solutions of the system (3.16)-(3.21) with corresponding pressure p b ′ and p b ′′ , and initial data (( υ ) , ( T ) ) and (( υ ) , ( T ) ) , respectively. Denote by v = υ − υ , p b = p b ′ − p b ′′ and T = T − T . Then we have ∂ t v − ∆ v − ∂ zz v + α − υ · ∇ v + α − ( v · ∇ ) υ + α − Φ( υ ) v z + α − Φ( v ) ∂ z υ + f k × v + α ∇ p b − αβ − R z − ∇ T dz ′ = 0 , (3.56) ∂ t T − ∆ T − ∂ zz T + α − υ · ∇ T + α − ( v · ∇ ) T + α − Φ( υ ) T z + α − Φ( v ) ∂ z T = 0 , (3.57) R − ∇ · vdz = 0 , (3.58) v ( x, y, z, t ) = ( υ ) − ( υ ) , T ( x, y, z, t ) = ( T ) − ( T ) , (3.59)( v, T ) satisfies the boundary value conditions (3 . − (3 . . (3.60) ultiplying L v in equation (3.56) and integrating with respect to spatial variable, we have12 ∂ t ( |∇ v | + | ∂ z v | ) + | ∆ v | + | ∂ z v | + |∇ v z | = − α − Z O ( υ · ∇ v ) · L v − α − Z O Φ( υ ) v z · L v − α − Z O (Φ( v ) ∂ z υ ) · L v − α − Z O [( v · ∇ ) υ ] · L v − Z O ( f k × v ) · L v + αβ − Z O ( Z z − ∇ T dz ′ ) · L v := K ( t ) + K ( t ) + K ( t ) + K ( t ) + K ( t ) + K ( t ) . (3.61)Using the Agmon inequality and the H¨older inequality, we obtain K ( t ) ≤ α − | v | ∞ |∇ v || L v |≤ Cα − k v k k v k k v kk v k ≤ Cα − k v kk v k k v k + ε k v k . Applying the H¨older inequality, the interpolation inequality and the Sobolev embedding theorem,we get K ( t ) ≤ α − Z O | Z − ∇ · v dz | · | v z | · | L v |≤ α − |∇ · ¯ v | L ( M ) Z − | v z | L ( M ) | L v | L ( M ) dz ≤ Cα − k v k k v k k v k k v k k v k ≤ ε k v k + Cα − k v k k v k k v k . Similar to the above, we obtain K ( t ) ≤ α − Z O | Z − ∇ · vdz | · | ∂ z v | · | L v |≤ α − |∇ · ¯ v | L ( M ) Z − | ∂ z v | L ( M ) | L v | L ( M ) dz ≤ Cα − k v k k v k Z − k v k H ( M ) k v k H ( M ) k v k dz ≤ ε k v k + Cα − k v k k v k k v k . With the help of the H¨older inequality, we deduce that K ( t ) ≤ α − | v | |∇ v | | L v | ≤ ε k v k + Cα − k v k k v k , and K ( t ) + K ( t ) ≤ ε k v k + C | v | + Cα β − k T k . By the boundary conditions, we know that | T | is small than | T z | + | T ( z = 0) | L ( M ) , then k T k isequivalent to |∇ T | + | T z | + | T ( z = 0) | L ( M ) . Keeping this in mind and taking an inner product f the equation (3.57) with L T , we have12 ∂ t ( |∇ T | + | T z | + γ | T ( z = 0) | ) + | ∆ T | + | T zz | + |∇ T z | + γ |∇ T ( z = 0) | = − α − Z O ( υ · ∇ T ) L T − α − Z O ( v · ∇ T ) L T − α − Z O Φ( υ ) T z L T + α − Φ( v ) ∂ z T L T := J ( t ) + J ( t ) + J ( t ) + J ( t ) . By the Agmon inequality, we get that J ( t ) + J ( t ) ≤ α − | v | ∞ |∇ T || L T | + α − | v | |∇ T | | L T |≤ Cα − k v k k v k k T kk T k + Cα − k v kk T k k T k ≤ ε k T k + Cα − k v k k T k + Cα − k v kk v k k T k . Taking an similar argument as K ( t ), we obtain J ( t ) ≤ α − Z O (cid:16) Z − |∇ · v | dz (cid:17) · | T z | · | L T |≤ α − (cid:16) Z − | L T | L ( M ) | T z | L ( M ) dz (cid:17) | Z − |∇ · v | dz | L ( M ) ≤ Cα − Z − k T k H ( M ) k T k H ( M ) dz Z − |∇ · v | L ( M ) dz ≤ Cα − k T k k T k Z − k v k H ( M ) k v k H ( M ) dz ≤ Cα − k T k k T k k v k k v k ≤ ε k T k + Cα − k T k k v k k v k . Similarly, we can prove that J ( t ) ≤ α − Z O ( Z − |∇ · v | ) | ∂ z T | · | L T |≤ α − (cid:16) Z − | L T | L ( M ) | ∂ z T | L ( M ) dz (cid:17) | Z − |∇ · v | dz | L ( M ) ≤ α − (cid:16) Z − k T k H ( M ) k T k H ( M ) k T k H ( M ) dz (cid:17) Z − |∇ · v | L ( M ) dz ≤ Cα − k T k k T k k T k k v k k v k ≤ ε k T k + ε k v k + Cα − k T k k T k k v k . Denote η ( t ) := k v ( t ) k + k T ( t ) k ,ξ ( t ) := α − k v kk v k + α − k v k k v k + α − k v k k v k + α k v k + α k T k + α − k T k k T k + 1 . Notice that |∇ v | + | ∂ z v | is equivalent to k v k and |∇ T | + | T z | + | T ( z = 0) | L ( M ) is equivalent to k T k , letting ε be small enough, we deduce from the above estimates of K − K and J − J that dη ( t ) dt + k v k + k T k ≤ η ( t ) ξ ( t ) . (3.62) ince ( v i ( t ) , T i ( t )) , i = 1 , , is the solution of stochastic PEs in sense of Definition 3.2, we have Z tt ξ ( s ) ds < ∞ , a.s., for all t ∈ ( t , ∞ ) , which implies that η ( t ) ≤ η ( t ) e C R tt ξ ( s ) ds . Therefore, we proved that for any t ∈ ( t , ∞ ) , ( u ( t ) , θ ( t )) is Lipschitz continuous in V × V withrespect to the initial data ( u ( t ) , θ ( t )) , which is equivalent to that strong solution ( υ ( t ) , T ( t )) of(1.9)-(1.14) is Lipschitz continuous in V × V with respect to the initial data ( v t , T t ) , for any t ∈ ( t , ∞ ) . (cid:3) Remark 3.1 (1)
In the above theorem, we have obtained the continuity of the strong solution with respect toinitial data in ( H ( O )) . This is a key to prove the compact property of the solution operator in V . Notice that the authors only proved the strong solution is Lipschitz continuous in the space ( L ( O )) with respect to the initial data in [16], which is not enough to obtain the asymptoticalbehavior in ( H ( O )) . (2) With the help of Lemma 3.1, we have established the the continuity of strong solution withrespect to time in V and a priori estimates to prove the compact property of the solution operatorin V . (3) We release the regularity of Q from H ( O ) to L ( O ) , which is more natural. In this section, we establish the existence of random attractor. Firstly, we recall some preliminariesfrom [3].Denote by C ( R ; X ) the space of continuous functions with values in X and equal to 0 at t = 0.Let ( X, d ) be a polish space and ( ˜Ω , ˜ F , ˜ P ) be a probability space, where ˜Ω is the two-sided Wienerspace C ( R ; X ). Definition 4.1
A family of maps S ( t, s ; ω ) : X → X, −∞ < s ≤ t < ∞ , parametrized by ω ∈ ˜Ω , is said to be a stochastic flow, if ˜ P -a.s., (i) S ( t, r ; ω ) S ( r, s ; ω ) x = S ( t, s ; ω ) x for all s ≤ r ≤ t, x ∈ X, (ii) S ( t, s ; ω ) is continuous in X, for all s ≤ t, (iii) for all s < t and x ∈ X , the mapping ω S ( t, s ; ω ) x is measurable from ( ˜Ω , ˜ F ) to ( X, B ( X )) where B ( X ) is the Borel σ -algebra of X , (iv) for all t, x ∈ X, the mapping s S ( t, s ; ω ) is right continuous at any point. Definition 4.2
A set-valued map K : ˜Ω → X taking values in the closed subsets of X is said tobe measurable if for each x ∈ X the map ω d ( x, K ( ω )) is measurable, where d ( A, B ) = sup { inf { d ( x, y ) : y ∈ B } : x ∈ A } for A, B ∈ X , A, B = ∅ , and d ( x, B ) = d ( { x } , B ) . Since d ( A, B ) = 0 if and only if A ⊂ B , d is not a metric. efinition 4.3 A closed set valued measurable map K : ˜Ω → X is called a random closed set. Definition 4.4
Given t ∈ R and ω ∈ ˜Ω , K ( t, ω ) ⊂ X is called an attracting set at time t if for allbounded sets B ⊂ X, d ( S ( t, s ; ω ) B, K ( t, ω )) → , provided s → −∞ . Moreover, if for all bounded sets B ⊂ X, there exists t B ( ω ) such that for all s ≤ t B ( ω ) S ( t, s ; ω ) B ⊂ K ( t, ω ) , we say K ( t, ω ) is an absorbing set at time t. Let { ϑ t : ˜Ω → ˜Ω } , t ∈ T = R , be a family of measure preserving transformations of the probabilityspace ( ˜Ω , ˜ F , ˜ P ) such that for all s < t and ω ∈ ˜Ω, the following (a) ( t, ω ) → ϑ t ω is measurable, (b) ϑ t ( ω )( s ) = ω ( t + s ) − ω ( t ), (c) S ( t, s ; ω ) x = S ( t − s, ϑ s ω ) x ,hold. Then, ( ϑ t ) t ∈ T is a flow and (( ˜Ω , ˜ F , ˜ P ) , ( ϑ t ) t ∈ T ) is a measurable dynamical system. Definition 4.5
Given a bounded set B ⊂ X , the set Ω( B, t, ω ) = \ T ≤ t [ s ≤ T S ( t, s, ω ) B is said to be the Ω -limit set of B at time t . Obviously, if denote Ω( B, , ω ) = Ω( B, ω ) , we have Ω( B, t, ω ) = Ω(
B, ϑ t ω ) . It’s easy to identifyΩ(
B, t, ω ) = n x ∈ X : there exists s n → −∞ and x n ∈ B such that lim n →∞ S ( t, s n , ω ) x n = x o . Furthermore, if there exists a compact attracting set K ( t, ω ) at time t, it is not difficult to checkthat Ω( B, t, ω ) is a nonempty compact subset of X and Ω( B, t, ω ) ⊂ K ( t, ω ) . Definition 4.6
For all t ∈ R and ω ∈ ˜Ω , a random closed set ω → A ( t, ω ) is called the randomattractor, if ˜ P − a.s., (1) A ( t, ω ) is a nonempty compact subset of X, (2) A ( t, ω ) is the minimal closed attracting set, i.e., if ˜ A ( t, ω ) is another closed attracting set, then A ( t, ω ) ⊂ ˜ A ( t, ω ) , (3) it is invariant, in the sense that, for all s ≤ t, S ( t, s ; ω ) A ( s, ω ) = A ( t, ω ) . Let A ( ω ) = A (0 , ω ), then the invariance property can be written as S ( t, s ; ω ) A ( ϑ s ω ) = A ( ϑ t ω ) . We will prove the existence of the random attractor using Theorem 2.2 in [3]. For the convenienceof reference, we cite it here. heorem 4.1 Let ( S ( t, s ; ω )) t ≥ s,ω ∈ ˜Ω be a stochastic dynamical system satisfying (i)-(iv) . Assumethat there exists a family of measure preserving mappings ϑ t , t ∈ R such that (a) - (c) hold and thereexists a compact attracting set K ( ω ) at time , for ˜ P -a.s.. Set A ( ω ) = [ B ⊂ X,B bounded Ω( B, ω ) where the union is taken over all the bounded subsets of X . Then we have ˜ P -a.s., (1) A ( ω ) is a nonempty compact subset of X . If X is connected, it is a connected subset of K ( ω ) . (2) The family A ( ω ) , ω ∈ Ω , is measurable. (3) A ( ω ) is invariant in the sense that S ( t, s ; ω ) A ( ϑ s ω ) = A ( ϑ t ω ) , s ≤ t. (4) It attracts all bounded sets from −∞ : for bounded B ⊂ X and ω ∈ ˜Ω d ( S ( t, s ; ω ) B, A ( ϑ t ω )) → , when s → −∞ . Moreover, it is the minimal closed set with this property: if ˜ A ( ϑ t ω ) is a closed attracting set,then A ( ϑ t ω ) ⊂ ˜ A ( ϑ t ω ) . (5) For any bounded set B ⊂ X, d ( S ( t, s ; ω ) B, A ( ϑ t ω )) → in probability when t → ∞ . And if the time shift ϑ t , t ∈ R is ergodic, (6) there exists a bounded set B ⊂ X such that A ( ω ) = A ( B, ω ) , (7) A ( ω ) is the largest compact measurable set which is invariant in sense of Definition 4.6. Before showing the existence of random attractor, we recall the Aubin-Lions Lemma, which isvital to the proof of Theorem 4.2.
Lemma 4.1
Let B , B, B be Banach spaces such that B , B are reflexive and B c ⊂ B ⊂ B . For < T < ∞ , set X := n h (cid:12)(cid:12)(cid:12) h ∈ L ([0 , T ]; B ) , dhdt ∈ L ([0 , T ]; B ) o . Then X is a Banach space equipped with the norm | h | L ([0 ,T ]; B ) + | h ′ | L ([0 ,T ]; B ) . Moreover, X c ⊂ L ([0 , T ]; B ) . The main result is:
Theorem 4.2
Let Q ∈ L ( O ) , υ ∈ V , T ∈ V . Then the solution operator ( S ( t, s ; ω )) t ≥ s,ω ∈ ˜Ω of3D stochastic PEs (1.9)-(1.14): S ( t, s ; ω )( υ s , T s ) = ( υ ( t ) , T ( t )) satisfies (i)-(iv) in Definition 4.1and possesses a compact absorbing ball B (0 , ω ) in V at time . Furthermore, for ˜ P -a.s. ω ∈ Ω , set A ( ω ) = [ B ⊂ V Ω( B, ω ) where the union is taken over all the bounded subsets of V . Then A ( ω ) is the random attractorof stochastic PEs (1.9)-(1.14) and possesses the properties (1)-(7) of Theorem 4.1 with space X replaced by space V. roof. Denote by w = ( w := P nk =1 α k w k , w := P nk =1 β k w k ) the R − valued Brownian motion,which has a version ω in C ( R , R ) := ˜Ω, the space of continuous functions which are zero at zero . In the following, we consider a canonical version of w given by the probability space( C ( R , R ) , B ( C ( R , R )) , ˜ P ), where ˜ P is the Wiener-measure generated by w and B ( C ( R , R )) isthe family of Borel subsets of C ( R , R ). Now, define the stochastic flow ( S ( t, s ; ω )) t ≥ s, ω ∈ ˜Ω by S ( t, s ; ω )( υ s , T s ) = ( α − ( t ) u ( t, ω ) , β − ( t ) θ ( t, ω )) , (4.63)where ( υ, T ) is the strong solution to (1.9)-(1.14) with ( υ s , T s ) = ( α − ( s ) u s ( s, ω ) , β − ( s ) θ s ( s, ω ))and ( u, θ ) is the strong solution to (3.16)-(3.21). It can be checked that assumptions (i)-(iv) and (a)-(c) of stochastic dynamics are satisfied with X = V . Indeed, properties (i),(ii),(iv) of thesolution operator ( S ( t, s ; ω )) t ≥ s,ω ∈ ˜Ω follows by Theorem 3.2 and property (iii) of the solutionoperator also holds with the help of the global existence of strong solution to (1.9)-(1.14) restupon Faedo-Galerkin method. Furthermore, ( ˜Ω , B ( C ( R , R )) , ˜ P , ϑ ) is an ergodic metricdynamical system.In the following, we will prove the existence of the random attractor. Let( u ( t, ω ; t , u ) , θ ( t, ω ; t , θ )) be the solution to (3 . − (3 .
21) with initial value u ( t ) = u and θ ( t ) = θ . By the law of the iterated logarithm, we havelim t →−∞ P nk =1 α k w k t = lim t →−∞ P nk =1 β k w k t = 0 . (4.64)Obviously, t → β ( t ) e λt is pathwise integrable over ( −∞ , λ is positive. And we havelim t →−∞ β ( t ) e λt = 0 , ˜ P − a.e.. (4.65)In view of (3.30), (3.31) and (4.65), for ˜ P − a.e. ω ∈ ˜Ω, there exists a random variable r ( ω ),depending only on λ such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and ( u , θ ) ∈ V with k u k + k θ k ≤ ρ, θ ( t, ω ; t , θ ) satisfiessup t ∈ [ − , | θ ( t, ω ; t , θ ) | + Z − k θ ( s ) k ds ≤ r ( ω ) . (4.66)In view of (3.33) and (4.66), taking a similar argument as (4.66), for ˜ P − a.e. ω ∈ ˜Ω, we deduce thatthere exists random variable r ( ω ), depending only on λ such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and ( u , θ ) ∈ V with k u k + k θ k ≤ ρ, u ( t, ω ; t , u ) satisfiessup t ∈ [ − , | u ( t, ω ; t , u ) | + Z − k u ( s ) k ds ≤ r ( ω ) . (4.67)By (3.40), repeating the argument as in (4.66), for ˜ P − a.e. ω ∈ ˜Ω, there exists random variable r ( ω ), depending only on λ such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and ( u , θ ) ∈ V with k u k + k θ k ≤ ρ, θ ( t, ω ; t , θ ) satisfiessup t ∈ [ − , | θ ( t, ω ; t , θ ) | ≤ r ( ω ) . (4.68)Integrating (3.44) with respect to time over [ t, −
3] yields, | ˜ u ( − | ≤ (cid:16) | ˜ u ( t ) | + C Z − t α ( s ) β − ( s ) | θ ( s ) | ds (cid:17) e C R − t α − ( s ) k u ( s ) k (1+ α − ( s ) | u ( s ) | ) ds , (4.69)Integrating (4.69) with respect to t over [ − , − | ˜ u ( − | ≤ (cid:16) Z − − | ˜ u ( t ) | ds + C Z − − α ( s ) β − ( s ) | θ ( s ) | ds (cid:17) × e C R − − α − ( s ) k u ( s ) k (1+ α − ( s ) | u ( s ) | ) ds . (4.70) herefore, by virtue of (4.66)-(4.68), we conclude that for ˜ P − a.e. ω ∈ ˜Ω, there exists randomvariable C ( ω ), depending only on λ such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and ( u , θ ) ∈ V with k u k + k θ k ≤ ρ, ˜ u ( − , ω ; t , u ) satisfies | ˜ u ( − | ≤ C ( ω ) . (4.71)Integrating (3.44) with respect to time over [ − , t ], we get | ˜ u ( t ) | ≤ (cid:16) | ˜ u ( − | + C Z t − α ( s ) β − ( s ) | θ ( s ) | ds (cid:17) e C R t − α − ( s ) k u ( s ) k (1+ α − ( s ) | u ( s ) | ) ds . (4.72)In view of (4.66)-(4.68) and (4.71), for ˜ P − a.e. ω ∈ ˜Ω, we deduce that there exists random variable r ( ω ) such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and( u , θ ) ∈ V with k u k + k θ k ≤ ρ, u ( t, ω ; t , u ) satisfiessup t ∈ [ − , | ˜ u ( t, ω ; t , ˜ u ) | ≤ r ( ω ) . (4.73)Taking integration of (3.43) with respect to time over [ − ,
0] yields, Z − ( |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | + | ˜ u |∇ ˜ u || + | ˜ u | ∂ z ˜ u || ) dt ≤ | ˜ u ( − | + C Z − α β − | θ | | ˜ u | + C Z − ( α − k u k + α − | u | k u k ) | ˜ u | . (4.74)By (4.67), (4.68) and (4.74), for ˜ P − a.e. ω ∈ ˜Ω, we infer that that there exists random variable C ( ω ) such that for ρ > t ( ω ) ≤ − t ≤ t ( ω ) and ( u , θ ) ∈ V with k u k + k θ k ≤ ρ, ˜ u ( t, ω ; t , u ) satisfies Z − ( |∇ ( | ˜ u | ) | + | ∂ z ( | ˜ u | ) | + | ˜ u |∇ ˜ u || + | ˜ u | ∂ z ˜ u || ) dt ≤ C ( ω ) . (4.75)By (3.46), (4.68) and (4.75), proceeding as (4.73), for ˜ P − a.e. ω ∈ ˜Ω, there exists random variable C ( ω ) such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and( u , θ ) ∈ V with k u k + k θ k ≤ ρ, ¯ u ( t, ω ; t , ¯ u ) satisfiessup t ∈ [ − , |∇ ¯ u ( t, ω ; t , ¯ u ) | ≤ C ( ω ) . (4.76)In view of (3.49) and (4.76), following the steps in (4.73), for ˜ P − a.e. ω ∈ ˜Ω, there exists a randomvariable r ( ω ), such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and( u , θ ) ∈ V with k u k + k θ k ≤ ρ, satisfiessup t ∈ [ − , | ∂ z u ( t, ω ; t , u ) | + Z − |∇ u z ( s, ω ; t , u ) | ds ≤ r ( ω ) . (4.77)Regarding (3.53), (4.66), (4.67) and (4.77), we repeat the procedures of deriving (4.73) and (4.74).For ˜ P − a.e. ω ∈ ˜Ω, there exists a random variable r ( ω ) such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and ( u , θ ) ∈ V with k u k + k θ k ≤ ρ, u ( t, ω ; t , u ) satisfiessup t ∈ [ − , |∇ u ( t, ω ; t , u ) | + Z − | ∆ u ( s, ω ; t , u ) | ds ≤ r ( ω ) . (4.78) y (3.55), (4.66), (4.77) and (4.78), proceeding as above, for ˜ P − a.e. ω ∈ ˜Ω, there exists a randomvariable r ( ω ) such that for arbitrary ρ > t ( ω ) ≤ − t ≤ t ( ω ) and( u , θ ) ∈ V with k u k + k θ k ≤ ρ, θ ( t, ω ; t , θ ) satisfiessup t ∈ [ − , k θ ( t, ω ; t , θ ) k ≤ r ( ω ) . Now we are ready to prove the desired compact result. Let r ( ω ) = r ( ω ) + r ( ω ) + r ( ω ), then B ( − , r ( ω )), the ball of center 0 ∈ V and radius r ( ω ) , is an absorbing set at time − S ( t, s ; ω )) t ≥ s,ω ∈ ˜Ω . According to Theorem 4.1, in order to prove the existence of the randomattractor in the space V , we need to to construct a compact absorbing set at time 0 in V . Let B be a bounded subset of V , set C T := n(cid:16) A υ, A T (cid:17)(cid:12)(cid:12)(cid:12) ( υ ( − , T ( − ∈ B , ( υ ( t ) , T ( t )) = S ( t, − ω )( υ ( − , T ( − , t ∈ [ − , o . We claim that C T is compact in L ([ − , H ) . Indeed, the space V × V ⊂ H × H is compactas V i ⊂ H i is compact. Let ( υ ( − , T ( − ∈ B , by the argument of step 2 in the proof ofTheorem 4.1, we have( A u, A θ ) ∈ L ([ − , V × V ) , ( ∂ t A u, ∂ t A θ ) ∈ L ([ − , V ′ × V ′ ) . Therefore, we deduce the result from Lemma 4.1 with B = V × V , B = H × H , B = V ′ × V ′ . Now, we aim to show that for any fixed t ∈ ( − , , ω ∈ ˜Ω , S ( t, − ω ) is a compact operator in V .Taking any bounded sequences { ( ν ,n , τ ,n ) } n ∈ N in B , for any fixed t ∈ ( − , ω ∈ ˜Ω , we devoteto extracting a convergent subsequence from { S ( t, − ω )( ν ,n , τ ,n ) } . Since { ( A υ, A T ) } ⊂ C T , by Lemma 4.1, there is a function ( ν ∗ , θ ∗ ) ∈ L ([ − , V ) and a subsequence of { S ( t, − ω )( ν ,n , τ ,n ) } n ∈ N still denoted by { S ( t, − ω )( ν ,n , τ ,n ) } n ∈ N such thatlim n →∞ Z − k S ( t, − ω )( ν ,n , τ ,n ) − ( ν ∗ ( t ) , θ ∗ ( t )) k dt = 0 . (4.79)By the measure theory, we know that the convergence in mean square implies almost sureconvergence. Therefore, it follows from (4.79) that there exists a subsequence { S ( t, − ω )( ν ,n , τ ,n ) } n ∈ N such thatlim n →∞ k S ( t, − ω )( ν ,n , τ ,n ) − ( ν ∗ ( t ) , θ ∗ ( t )) k = 0 , a.e. t ∈ ( − , . (4.80)Fix any t ∈ ( − , . By (4.80), we can select a t ∈ ( − , t ) such thatlim n →∞ k S ( t , − , ω )( ν ,n , τ ,n ) − ( ν ∗ ( t ) , θ ∗ ( t )) k = 0 . Then by the continuity of S ( t − t , t ; ω ) in V with respect to the initial value, we have S ( t, − ω )( ν ,n , τ ,n ) = S ( t − t , t ; ω ) S ( t , − ω )( ν ,n , τ ,n ) → S ( t − t , t ; ω )( ν ∗ ( t ) , θ ∗ ( t )) , in V. Hence, for any t ∈ ( − , { S ( t, − ω )( ν ,n , τ ,n ) } n ∈ N in V, which implies that for any fixed t ∈ ( − , , ω ∈ ˜Ω , S ( t, − ω ) is acompact operator in V. Set B (0 , ω ) = S (0 , − ω ) B ( − , r ( ω )) , then, B (0 , ω ) is a closed set of S (0 , − ω ) B ( − , r ( ω )) in V. Using the above argument, we knowthat B (0 , ω ) is a random compact set in V. Precisely, B (0 , ω ) is a compact absorbing set in V attime 0 . Indeed, for ( ν ,n , τ ,n ) ∈ B , there exists s ( B ) ∈ R − such that for any s ≤ s ( B ) , we have S (0 , s ; ω )( ν ,n , τ ,n ) = S (0 , − ω ) S ( − , s ; ω )( ν ,n , τ ,n ) ⊂ S (0 , − ω ) B ( − , r ( ω )) ⊂ B (0 , ω ) . Therefore, we conclude the result from Theorem 4.1. (cid:3) Existence of invariant measure
Up to now, we are ready to prove the existence of invariant measure of the system (1.9)-(1.14).Let U = ( v , T ) ∈ V , U ( t, ω ; U ) := ( v ( t, ω ; t , v ) , T ( t, ω ; t , T )) is the solution to (3.16)-(3.21)with the initial value U . Following the standard argument, we can show that U ( t, ω ; U ) , t ∈ [ t , T ]is Markov process in the sense that for every bounded, B ( V )-measurable F : V → R , and all s, t ∈ [ t , T ], t ≤ s ≤ t ≤ T , E ( F ( U ( t, ω ; U )) |F s )( ω ) = E ( F ( U ( t, s, U ( s )))) for ˜ P − a.e. ω ∈ Ω , where F s = F t ,s (see (3.15)), U ( t, s, U ( s )) is the solution to (1.9)-(1.14) at time t with initial data U ( s ) . For B ∈ B ( V ), define ˜ P t ( U , B ) = ˜ P (( U ( t, ω ; U ) ∈ B ) . For any probability measure ν defined on B ( V ) , denote the distribution at time t of the solutionto (1.9)-(1.14) with initial distribution ν by( ν ˜ P t )( · ) = Z V ˜ P t ( x, · ) ν ( dx ) . For t ≥ t and any continuous and bounded function f ∈ C b ( V ; R ), we have˜ P t f ( U ) = E [ f ( U ( t, ω ; U )] = Z V f ( x )˜ P t ( U , dx ) . Definition 5.1
Let ρ be a probability measure on B ( V ) . ρ is called an invariant measure for ˜ P t , if Z V f ( x ) ρ ( dx ) = Z V ˜ P t f ( x ) ρ ( dx ) for all f ∈ C b ( V ; R ) and t ≥ . Let µ · be a transition probability from ˜Ω to V , i.e., µ · is a Borel probability measure on V and ω → µ · ( B ) is measurable for every Borel set B ⊂ V. Denote by P ˜Ω ( V ) the set of transitionprobabilities with µ · and ν · identified if ˜ P { ω : µ ω = ν ω } = 0 . In view of Proposition 4 . µ · ∈ P ˜Ω ( V ) for S such that µ ω ( A ( ω )) = 1,˜ P -a.e.. Therefore, referring to [2], there exists an invariant measure for the markov semigroup ˜ P t and it is given by ρ ( B ) = Z ˜Ω µ ω ( B )˜ P ( dω ) , where B ⊆ V is a Borel set. If the invariant measure ρ for ˜ P is unique, the invariant Markovmeasure µ · for S is unique and given by µ ω = lim t →∞ S (0 , − t, ω ) ρ. Based on the above, we arrive at
Theorem 5.1
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