# Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises

Authors: Jianhua Huang; Yuhong Li; Jinqiao Duan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is devoted to the investigation of random dynamics of the stochastic Boussinesq equations driven by Lévy noise. Some fundamental properties of a subordinator Lévy process and the stochastic integral with respect to a Lévy process are discussed, and then the existence, uniqueness, regularity, and the random dynamical system generated by the stochastic Boussinesq equations are established. Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.


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## Body

## 1. Introduction

Dynamical systems driven by non-Gaussian processes, such as Lévy processes, have attracted a lot of attention recently. Ordinary differential equations driven by Lévy processes have been summarized in [1]. Peszat and Zabczyk [2] have presented a basic framework for partial differential equations driven by Lévy processes.The Navier–Stokes fluid equations are often coupled with other equations, especially, with the scalar transport equations for fluid density, salinity, or temperature. These coupled equations model a variety of phenomena arising in environmental, geophysical, and climate systems. The related Boussinesq fluid equations [3–5] under Gaussian fluctuations have been recently studied, for example, existence and uniqueness of solutions [6], stochastic flow, dynamical impact under random dynamical boundary conditions [7, 8], and large deviation principles [9, 10], among others.Motivated by a recent work on a simple stochastic partial differential equation with Lévy noise [11], we study the stochastic Boussinesq equations driven by some special Lévy noises, and we consider the random dynamics of this stochastic system. Specifically, for a given bounded C1-smooth domain D⊂ℝ2 with sufficient smooth boundary, we consider the following stochastic Boussinesq equations driven by subordinator Lévy noise:
(1)dudt=(1ReΔu-∇p-u·∇u-1Fr2θe2)+dY1(t),onD×ℝ+,dθdt=(1RePrΔθ-u·∇θ)+dY2(t),onD×ℝ+,divu=0,onD×ℝ+,u(0)=u0,θ(0)=θ0,
where u=u(x,t)=(u1,u2)∈ℝ2 is the velocity vector, θ=θ(t,x)∈ℝ is salinity, p(t,x)∈ℝ is the pressure term, x=(ξ,η)∈D⊂ℝ2,  Δ denotes the Laplacian operator, and ∇ denotes the gradient operator. Moreover, Fr is the Froude number, Re is the Reynolds number, Pr is the Prandtl number, and e2∈ℝ2 is a unit vector in the upward vertical direction. The initial data u0, θ0 are given. Both Y1(t) and Y2(t) are subordinator Lévy processes on Hilbert spaces H1 and H2, which will be specified later. The present paper is devoted to the existence, uniqueness, regularity, and the cocycle property of solution for stochastic Boussinesq equations (1).This paper is organized as follows. In Section2, we first present some properties of the subordinator Lévy process Y(t), then review some fundamental properties of the stochastic integral with respect to Lévy process Y(t). Section 3 is devoted to the existence, uniqueness, regularity, and the cocycle property of the stochastic Boussinesq equations. Finally, some discussions on the global weak solution of stochastic Boussinesq equations driven by general Lévy noise are also presented in Section 4.


## 2. Preliminaries

In this section, we introduce some operators and fraction spaces and then present some properties of the subordinator Lévy processY(t) and the stochastic integral with respect to Lévy process Y(t).In order to reformulate the stochastic Boussinesq equations (1) as an abstract stochastic evolution, we introduce the following function spaces.DenoteL2(D) to be the space of functions defined on D, which are L2-integrable with respect to the Lebesgue measure dx=dx1dx2, endowed with the usual scalar product and norm, that is, for u,v∈L2(D),
(2)(u,v)=∫D‍u(x)v(x)dx,|u|={(u,u)}1/2.Form∈ℤ+∪{0} and q∈(1,∞), define
(3)Hm,q(D)={u∈Lq(D):Dαu∈Lq(D),α∈ℕ2,0≤|α|≤m}
as the usual Soblev space with scalar product
(4)(u,v)m=∑0≤|α|≤m‍(Dαu,Dαv)Lq(D)
and the induced norm
(5)|u|m=∥u∥Hm(D)=(∑0≤|α|≤m|Dαu|Lq(D)q)1/2,
where Dαu is the αth order weak derivative of u.Fors∈ℝ, let Hs,q(D) be defined by the complex interpolation method [12] as follows.
(6)Hβ,q(D)=[Hk,q(D),Hm,q(D)]θ,
where k,m∈ℕ, θ∈(0,1), and k<m are chosen to satisfy
(7)β=(1-θ)k+θm.
The closure of C0∞(D) in the Banach space Hs,q(D), s≥0, q∈(1,∞), will be denoted by H0s,q(D).The following product spaces are needed:(8)𝒱={u=(u1,u2)∈(C∞(D))2×C∞(D),∇·u=0},𝕃q(D)=(Lq(D))2×Lq(D),ℍs,q(D)=(Hs,q(D))2×Hs,q(D)={u=(u1,u2)∈(Hs,q(D))2×Hs,q(D),∇·u=0(Hs,q(D))2},𝕍s,q(D)={u=(u1,u2)∈ℍ0s,q(D),∇·u=0}.LetHs,q(D) denote the closure of 𝒱 with respect to the ℍs,q-norm, Vs,q(D) denote the closure of 𝒱 with respect to the 𝕍s,q-norm, and V′ be the dual space of Vs,q(D). In particular, we denote by H1,2 and V1,2  H and V, respectively.Denote(9)A1u=Δu(t),A2θ=Δθ(t),B1(u1,u2)=(u1·∇)u2,B2(u1,θ2)=(u1·∇)θ2,U0=(u0θ0)∈H,U(t)=(u(t)θ(t))∈V,R(U)=(-1Fr2θe2,0),Y(t)=(Y1(t)Y2(t))∈H=H1×H2,
where ν=1/Re and k=1/RePr.Now, we define the following two operators:(10)A:V⟶V′:V∋U=(u,θ)⟼AU=(νA1ukA2θ),B:V×V⟶V′:V×V∋(U1,U2)⟼B(U1,U2)=(B1(u1,u2)B2(u1,θ2)).Then, the stochastic Boussinesq system (1) can be rewritten as the following abstract stochastic evolution equation:
(11)dU(t)+[AU(t)+B(U(t),U(t))+R(U(t))]dt=dY(t),U(0)=U0.In order to apply the technique of the reproducing Kernel Hilbert space, it is better to introduce the definitionγ-radonifying.Definition 1 (see [13]).
LetK and X be Banach spaces, a bounded linear operator L:K→X is called γ-radonifying if and only if L(γK) is σ-additive, where γK is the canonical cylindrical finitely additive set-valued function (also called a Gaussian distribution) on K.The following is our standing assumption:Assumption 1.
SpaceE⊂H∩𝕃4 is a Hilbert space such that for some δ∈(0,1/2),
(12)A-δ:E⟶H∩𝕃4isγ-radonifying.Remark 2.
Under the above assumption, we have the facts thatE⊂H and the Banach space U is taken as H∩𝕃4 (see [11, 14, 15] for more details and related results). In fact, space E is the reproducing kernel Hilbert space of noise W(t) on H∩𝕃4.It is well known that subordinators form the subclass of increasing Lévy processes, which can be thought of as a random model of time evolution (see [16]). We will present some properties of the subordinator Lévy process Y(t), t≥0, then review briefly the stochastic integral with respect to Lévy process Y(t).Definition 3 (see [1, 2, 11]).
LetE be a Banach space, and let Y=(Y(t),t≥0) be an E-valued stochastic process defined on a probability space (Ω,ℱ,ℙ). Stochastic process Y is called a Lévy process if (L1)
Y(0)=0, a.s.;(L2)
processY has independent and stationary increments; and(L3)
processY is stochastically continuous, that is, for all δ>0 and for all s≥0,
(13)limt→sℙ(|Y(t)-Y(s)|>δ)=0.
A subordinator Lévy process is an increasing one-dimensional Lévy process.
Forp>0, Sub(p) denotes the set of all subordinator Lévy processes Z, whose intensity measure ρ satisfies the condition ∫01‍ηp/2ρ(dη)<∞.In the most interesting cases, the spaceE is a subspace of H, that is, E⊂H, and
(14)Y(t)=W(Z(t)),t≥0,
where Z=(Z(t))t≥0 is an independent subordinator process belonging to class Sub(p), p∈(1,2], W=(W(t))t≥0 is an H-valued cylindrical Wiener process defined on some Banach space U.We decompose theH-valued Lévy process Y(t) into two parts N1(t) and N2(t), the first one with small jumps and the second one with (relatively) large jumps, that is,
(15)Y(t)=N1(t)+N2(t),t≥0,
with ν being the intensity measure of Lévy process Y, N1 being the Lévy process with the intensity measure:
(16)ν1(Γ)=ν(Γ∩BU(0,1)),Γ∈ℬ(U),BU(0,1)denotestheunitballinU,
and N2 be the Lévy process with the intensity measure ν2=ν-ν1. Then N2 can be defined as a compound Poisson process with the intensity measure ν2, and N1, N2 can be defined by the Poisson random measure π which is defined as follows:
(17)π([0,1]×Γ)=Σs≤t1ΓΔY(s),Γ∈ℬ(U),
where ΔY(s)=Y(s+)-Y(s-), s≥0. Here, the symbol Δ denotes the increment of Y.We assume that the processY is right-continuous with left-hand side limits. Thus
(18)ΔY(s)=Y(s)-Y(s-),s≥0.Notice that asπ is a time homogenous Poisson random measure, Y can be expressed as
(19)Y(t)=Σs≤tΔY(s)=∫0t‍∫U‍uπ(dy,ds),t≥0.
Hence,
(20)N1(t)=Σs≤t1|ΔY(s)|<1ΔY(s)=∫0t‍∫|u|<1‍uπ(dy,ds),N2(t)=Σs≤t1|ΔY(s)|≥1ΔY(s)=∫0t‍∫|u|≥1‍uπ(dy,ds).Assume that the operatorΨ(t), t∈[0,T], is a strongly measurable function taking values in the space of all bounded linear operator from U to E. Let 0<τ1<τ2<τ3<⋯→∞ be the jump times for N2 and ΔN2(τk)=ΔY(τk)=Y(τk)-Y(τk-), k=1,2,…. Then, the stochastic integral of Ψ(t) with respect to jump process N2(t), t≥0, can be defined as
(21)∫0t‍Ψ(s)dN2(s)=Στk≤tΨ(τk)ΔN2(τk).Since the operatorΨ is taking values in E, it follows from the decomposition of Y that the sum of sequences is finite. Hence the stochastic integral of the operator Ψ with respect to N2 is taking values in E. Moreover, the stochastic integral of the operator Ψ(t), t∈[0,T], with respect to Lévy process Y can be defined by
(22)∫0t‍Ψ(s)dY(s)=∫0t‍Ψ(s)dN1(s)+∫0t‍Ψ(s)dN2(s)
and takes values in E as well (see [11] for more details).Next, we recall some basic definitions and properties for general random dynamical systems, which are taken from [7]. Let (H,d) be a complete separable metric space and (Ω,ℱ,ℙ) be a probability space.Definition 4.
(
Ω
,
ℱ
,
ℙ
,
(
θ
)
t
∈
ℝ
) is called a metric dynamical system if the mapping θ:ℝ×Ω→Ω is (ℬ(ℝ)×ℱ,ℱ) measurable, θ0=I, θs+t=θs∘θt for all t,s∈R, and θtℙ=ℙ for all t∈ℝ.Definition 5.
A random dynamical system (RDS) with timeT on (H,d) over {θt} on (Ω,ℱ,ℙ,(θt)t∈R) is a (ℬ(R+)×𝔉×ℬ(H),B(H))-measurable map:
(23)Π:T×H×Ω⟶H×Ω,Π(t,s,ω)=(S(t,ω)x,θtω)
such that (i)
S(0,ω)=Id (identity on H) for any ω∈Ω,(ii)
(Cocycle property) S(t+s,ω)=S(t,θsω)∘S(s,ω) for all s,t∈T and ω∈Ω.An RDS is said to be continuous or differentiable if for allt∈T, and an arbitrary outside outside ℙ-nullset B⊂Ω, w∈B the map S(t,ω):H→H is continuous or differentiable, respectively.Assume that the bounded linear operatorA generates a C0-semigroup S=(etA)t≥0 on a Hilbert space E and Y defined on a filtered probability space (Ω,ℱ,(ℱ)t≥0,ℙ) is a subordinator Lévy process taking values in a Hilbert space U.Consider the following stochastic Langevin equation:(24)dX(t)=AX(t)dt+dY(t),t≥t0,X(t0)=x∈E.Definition 6.
Letx∈E be a square integrable ℱt0-measurable random variable in E. A predicable process X:[t0,∞)×Ω→E is called a mild solution of the Langevin equation (24) with initial data (t0,x) if it is an adapted E-valued stochastic process and satisfies
(25)X(t)=S(t-t0)x+∫t0t‍S(t-s)dY(s),t≥t0.It is well known that the Ornstein-Uhlenbeck processX(t), t≥0, has some important integrability. Here we need the Banach space to be of type p, for p∈(1,2]. First we recall the definition briefly (see [14] for more details).Definition 7 (see [14]).
Forp∈(1,2], the Banach space E is called as type p, if and only if there exists a constant Kp(E)>0 such that for any finite sequence of symmetric independent identically distribution random variables ε1,…,εn:Ω→[-1,1], n∈ℕ, and any finite sequence x1,…,xn from E, satisfying
(26)𝔼|∑i=1n‍εixi|p≤Kp(E)∑i=1n‍|xi|p.
Moreover, if there exists a constantLp(E)>0 such that for every E-valued martingale {Mn}n=0N, N∈ℕ, satisfying
(27)supn𝔼|Mn|p≤Lp(E)∑n=0N‍𝔼|Mn-Mn-1|p,M-1=0,
the separable Banach space E is called a separable martingale type p-Banach space.Lemma 8 (see [11, Corollary 8.1, Proposition 8.4]).
Assume thatp∈(1,2], Z is a subordinator Lévy process from the class Sub(p), E is a separable type p-Banach, U is a separable Hilbert space U, E⊂U, and W=(W(t))t≥0 is a U-valued Wiener process.
Define theU-valued Lévy process as
(28)Y(t)=W(Z(t)),t≥0,
and define the process as
(29)X(t)=∫0t‍e(t-s)AdY(s).
Then, with probability 1, for all T>0,
(30)∫0T‍|X(t)|Epdt<∞,∫0T‍|X(t)|L44dt<∞.We have the following existence and regularity results, which have been studied in [2, 11].Theorem 9.
Assume thatE=U, S=S(t), t≥0 is the C0 semigroup generated by the bounded linear operator A in the space E. Then, if one of the following conditions is satisfied: (i)
p∈(0,1] or(ii)
p∈(1,2] and the Banach space E is of separable martingale type p-Banach space, the Langevin equation (24) admits one mild solution X(t)∈E, t>0. Moreover, if p∈(1,2], S=S(t), t≥0, is a C0-group in the separable martingale type p-Banach space E, then the mild solution X of the Langevin equation is a cádlág (right-continuous with left-hand side limits) process.Proof.
AsS=S(t), t≥0, is a C0-group in the separable martingale type p-Banach space E, the Hilbert space H is the reproducing kernel Hilbert space of W(1), and the embedding operator i:H↪E satisfies the γ-radonifying property. The proof of Theorem 9 is just a simple application of Theorems  4.1 and 4.4 in [11].


## 3. Cocycle Property of the Stochastic Boussinesq Equations

In this section, we will show the existence, uniqueness, regularity, and the cocycle property of the stochastic Boussinesq equations (11).It is well known that bothA1 and A2 are positive definite, self-adjoint operators, and denote D(A1) and D(A2) to be the domains of A1 and A2, respectively. Hence, the domain of the operator A can be represented as D(A)=D(A1)×D(A2). It follows from Lemma  2.2 in [7] that there exists positive numbers μ1,μ2, such that
(31)(A1u,u)≥μ1∥u∥(L2)22,(A2(u,θ),(u,θ))≥μ2∥(u,θ)∥2.
Let λ=min(μ1,μ2). Then
(32)(AU,U)≥λ∥U∥2.For any arbitraryU,V,W∈𝕍, we can define the following trilinear form b:U×V×W→ℝ by
(33)b(u,v,w)=〈B(u,v),w〉,b(U,V,W)=b1(u,v,w)+b2(u,v~,w~),b1(u,v,w)=∫D‍σi,j2ui∂vj∂xiwjdx,b2(u,v~,w~)=∫D‍σi2ui∂v~j∂xiw~jdx.
We have the following results.Lemma 10 (see [7, Lemma  2.3]).
IfU,V,W∈𝕍, then(34)b(U,V,W)=-b(U,W,V),(B(V,U),U)=b(V,U,U)=0.Lemma 11 (see [7, Lemma  2.4]).
There exists a constantcB>0 such that if u∈V1, θ,η∈V2, ϕ=(u,θ), then(1)
|b1(u,v,w)|≤cB∥u∥H1∥v∥H2∥w∥, u∈V, v∈D(A), w∈H,(2)
|b1(u,v,w)|≤cB∥u∥L21/2∥u∥H11/2∥v∥H2∥w∥1/2∥w∥H11/2, u∈V, v∈D(A), w∈V,(3)
|b1(u,v,u)|≤cB∥u∥H1∥v∥H1∥v∥, u∈V, v∈V,(4)
|b2(u,θ,w)|≤cB∥u∥1/2∥u∥H11/2∥θ∥H1∥w∥1/2∥w∥H11/2, u∈V, θ∈V, w∈V,(5)
|b2(u,θ,w)|≤cB∥u∥∥θ∥H2∥w∥H1, u∈H, θ∈D(A), w∈V.Definition 12.
AnH-valued (ℱt)t≥0 adapted and H4,2(D)-valued cádlág process u(t)(t≥0) is considered as a solution to (11), if for each T>0,
(35)sup0≤t≤T|U(t)|H2+∫0T‍|U(t)|𝕃4(D)4dt<∞,a.s.,
and for any ψ∈V∩H2,2(D), and for any t>0, ℙ-a.s.,
(36)(U(t),ψ)-(U0,ψ)-∫0t‍(U(s),Δψ)ds+∫0t‍(B(U,U),ψ(s))ds+∫0t‍(R(U),ψ)ds=(ψ,Y(t)).Denote(37)ℋ1,2(0,T)={thespaceofallfunctionsv∈L2(0,T;V)∩ℍ2,2(D)satisfyingv′∈L2(0,T;V′)}.Lemma 13.
Assume thatz∈L4(0,T;𝕃4(D)), g∈L2(0,T,V′), and v0∈H. Then there exists a unique v∈ℋ1,2(0,T) such that
(38)dvdt+Av+B(v,z)+B(z,v)+B(v,v)=g,t≥0,v(0)=v0.
Moreover,
(39)supt∈[0,T]|v(t)|2≤K2L2,∫0T‍|∇v(t)|2dt≤M2,∫0T‍|v′(t)|V′2dt≤N2,∫0T‍|v(t)|𝕃4(D)4dt≤2T1/2K3L3M,
where
(40)K2=e2∫0T‍|z(s)|𝕃44ds,L2=|v0|2+2∫0T‍|g(s)|V′2ds,M2=|v0|2+9KL∫0T‍|z(t)|L4(0,T,𝕃4(D))2+T1/42K3/2L1/2,
and the mapping L2(0,T,V′)×H∋(g0,v0)↦v∈ℋ1,2(0,T) is analytic.Proof.
It can be shown by the same approach as the one in Proposition  8.7 in [11].Lemma 14 (see [2, Proposition  10.1]).
Letu:[0,T]→B be a continuous function whose left derivative
(41)d-udt(t0)=limϵ→0,ϵ<0u(t0+ϵ)-u(t0)ϵ
exists at t0∈[0,T]. Then the function γ(t)=|u(t)|B, t∈[0,T], is left differentiable at t0 and
(42)d-γdt(t0)=min{〈x*,d-udt(t0)〉:x*∈∂|u(t0)|B}.In order to apply the Yosida approximation for the solution of (11), we need to introduce some definitions of dissipative mapping (operator) (see [17] for details).Definition 15.
Let(B,|·|B) be a separable Banach space, B* be the dual space of B. The subdifferential ∂|x|B of norm |·|B at x∈B is defined by the formula
(43)∂|x|B:={x*∈B*:|x+y|B-|x|B≥(x*,y),∀y∈B}.
A mappingF:D(F)⊂B→B is said to be dissipative, if for any x,y∈D(F), there exists z*∈∂|x-y|B such that
(44)〈z*,F(x)-F(y)〉≤0.
A dissipative mappingF:D(F)⊂B→B is called an m-dissipative mapping or maximal dissipative if the image of I-λF is equal to the whole space B for some λ>0 (and then for any λ>0), that is,
(45)range(I-λA)=B,forsomeλ>0.
Assume thatF is an m-dissipative mapping. Then its resolvent Jα and respectively the Yosida approximations Fα, α>0, are defined by
(46)Jαx=(I-αF)-1x∈domF,Fαx=1α(Jαx-x),∀x∈domJα=range(I-αF).Lemma 16 (see [2, Proposition  10.2]).
LetF:D(F)→B be an m-dissipative mapping on B. Then (1)
for allα>0 and x,y∈B, |Jα(x)-Jα(y)|B≤|x-y|B;(2)
the mappingFα, α>0, are dissipative and Lipschitz continuous:
(47)|Fα(x)-Fα(y)|B≤2α|x-y|B,∀x,y∈B.Moreover,|Fα(x)|B≤|F(x)|B, for all x∈D(F); and(3)
limα→0Fα(x)=x, for all x∈D(F)¯.The following theorem is one of the main results of this paper, which will be proved by applying the well-known Yosida approach.Theorem 17.
For everyu0∈H, under Assumption 1, the stochastic Boussinesq system (11) admits a unique cádlág mild solution u(t).Proof.
DenoteZA(ω) to be the stationary solution of Langevin equation (24). Let V=U-ZA. Then (11) is converted into the following evolution equation with random coefficients:
(48)dV=[AV+B(V+ZA,V+ZA)+R(V+ZA)]dt,t≥0,V(0)=U0,
where (A,D(A)) generates an analytic C0-semigroup S (see Section  2.2 in [2]). It follows from the proof of Theorem  10.1 in [2] that, for α>0, β>0, and sufficiently small η, the mappings A+η and B(·,·)+R(·)+η are m-dissipative. Hence, the Yosida approximations of the m-dissipative mappings A+η and B(·,·)+R(·)+η can be respectively denoted by
(49)(A+η)β=1β((I-β(A+η))-1-I),((B+R)+η)α=1α((I-α((B+R)+η))-1-I).
Now consider the following random approximate equation:(50)d-dtYα,β(t)=(A+η)βYα,β+(B+R+η)α(Yα,β+ZA(t-))-2ηYα,β-ηZA(t-),Yα,β(0)=U0.
It is easy to verify that ((A+η)β,D((A+η)β)) generates an analytic C0-semigroup Sβ. Notice that the Yosida approximate operators are Lipschitz. Therefore the random approximation equation (50) has a unique continuous solution Yα,β.
Next we will show that(51)limα→0[limβ→0Yα,β(t)]=Y(t),t≥0,
in H, and this limit is actually the mild solution of stochastic Boussinesq equation (48).
For the sake of simplification, we just present the estimations whenη=0, and the remaining estimates can be obtained by the similar arguments for η≠0.
LetYα be the solution of the integral equation:
(52)Yα(t)=S(t)U0+∫0t‍S(t-s)×(B(Yα(s)+ZA(s-),Yα(s)+ZA(s-))+R(Yα(s)+ZA(s-)))αds.
Notice that the operator (B(·,·)+R(·))α is Lipschitz continuous and ZA is cádlág. Hence, there exists a solution of random approximate equation (50), which is continuous in H.
Forα>0 and β>0, direct computation implies
(53)Yα-Yα,β=S(t)U0-Sβ(t)+∫0t‍[S(t-s)-Sβ(t-s)]×[B(Yα(s)+ZA(s-),Yα(s)+ZA(s-))+R(Yα(s)+ZA(s-))]αds+∫0t‍[Sβ(t-s)]×[[R(Yα,β(s)+ZA(s-))]α[B(Yα(s)+ZA(s-),Yα(s)+ZA(s-))+R(Yα(s)+ZA(s-))]α-[B(Yα,β(s)+ZA(s-),Yα,β(s)+ZA(s-)Yα,β(s))+R(Yα,β(s)+ZA(s-))]α]ds.
Since both A and B+R are m-dissipative. Therefore, there exists constant M, ω, and Cα such that for all t≥0, V,W∈H,
(54)∥Sβ(t)∥L(H,H)≤Meωt,|[B(V)+R(V)]α-[B(U)+R(U)]α|≤Cα|V-U|H.
Then
(55)|Yα(t)-Yα,β(t)|≤|S(t)U0-Sβ(t)U0|+MCα∫0t‍eω(t-s)|Yα(s)-Yα,β(s)|ds+∫0t‍|[Sβ(t-s)-S(t-s)]×[B(Yα,β(s)+ZA(s-),Yα,β(s)+ZA(s-))+R(Yα,β(s)+ZA(s-))]α|ds.
By the Hille-Yosida theorem, it follows that
(56)Sβ(t)U0⟶S(t)U0,asβ⟶0
uniformly in t on compact subsets U0 of H.
Hence, it follows that(57)|Yα(t)-Yα,β(t)|≤MCα∫0t‍|Yα(s)-Yα,β(s)|ds
uniformly on bounded intervals as β→0.
ByGronwall inequality, we have
(58)limβ→0supt≤T|Yα(t)-Yα,β(t)|=0,∀T<∞.
By Lemma14,
(59)d-dt|Yα,β(t)|=min{〈x*,d-dtYα,β(t)〉:x*∈∂|Yα,β(t)|}=min{〈x*,AβYα,β(t)+[B(Yα,β(s)+ZA(s-),Yα,β(s)+ZA(s-))+R(Yα,β(s)+ZA(s-))]α〉:x*∈∂|Yα,β(t)|}.
Recalling that both Aβ and [B(·,·)+R(·)] are m-dissipative and Aβ is linear, we obtain
(60)d-dt|Yα,β(t)|≤|[R(ZA(t-))]α[B(Yα,β(t)+ZA(t-),Yα,β(s)+ZA(t-))+R(Yα,β(t)+ZA(t-))]α|≤|B(Yα,β(t)+ZA(t-),Yα,β(s)+ZA(t-))+R(Yα,β(t)+ZA(t-))|,
that is,
(61)|Yα,β(t)|≤|U0|+∫0t‍|B(Yα,β(s)+ZA(s-),Yα,β(s)+ZA(s-))+R(Yα,β(s)+ZA(s-))|ds,t≥0.
It follows from the estimate (58) that, for any α>0, and t∈[0,T],
(62)|Yα(t)|≤|U0|+∫0t‍|[B(Yα,β(s)+ZA(s-),Yα,β(s)+ZA(s-))+R(Yα,β(s)+ZA(s-))|ds.
Similarly, by Lemma16, for t∈[0,T],
(63)12d-dt|Yα,β(t)-Yγ,β(t)|2=〈d-dt(Yα,β(t)-Yγ,β(t)),Yα,β(t)-Yγ,β(t)〉=〈(AβYα,β(t)-AβYγ,β(t))+[(B+R)α]×(Yα,β(t)+ZA(ω)(t-))-[(B+R)γ]×(Yγ,β(t)+ZA(ω)(t-)),Yα,β(t)-Yγ,β(t)〉≤〈[(B+R)α](Yα,β(t)+ZA(ω)(t-))-[(B+R)γ]×(Yγ,β(t)+ZA(ω)(t-)),Yα,β(t)-Yγ,β(t)〉≤(γ+α)[|(B+R)α(Yα,β(t)+ZA(ω)(t-))|+|(B+R)γ(Yγ,β(t)+ZA(ω)(t-))|]2≤(γ+α)[|(B+R)(Yα,β(t)+ZA(ω)(t-))|+|(B+R)(Yγ,β(t)+ZA(ω)(t-))|]2.
By the dissipation of the operators A, B, and R and estimates (63), there exists a constant C>0 such that
(64)12d-dt|Yα,β(t)-Yγ,β(t)|2≤C(α+γ),t∈[0,T].
Therefore
(65)|Yα,β(t)-Yγ,β(t)|2≤2C(α+γ)T,t∈[0,T].
By the estimate (58),
(66)|Yα(t)-Yγ(t)|2≤2C(α+γ)T,t∈[0,T].
Thus, Yα(t)→Y(t) in H uniformly on [0,T] as α→0.
Next, we are going to show that the solutionYα of the Yosida approximations equation is a mild solution:
(67)Yα(t)=S(t)U0+∫0t‍S(t-s)(B+R)α(Yα(s)+ZA(s))ds,t∈[0,T].
By the reflexivity of H1 and the estimate ∥Yα(t)∥H1≤C2∥U0∥H1, t∈[0,T], α>0, there exists a subsequence {Yα,n}, which converges weakly in H1 and weakly converges to the function Y(t) in H1. Since {Yα,n(t)} is strong convergent in L2, and
(68)∥Y(t)∥H1≤C2∥U0∥H1,t∈[0,T].
Let h∈L2, then
(69)〈Yα(t),h〉L2=〈S(t)U0,h〉L2+∫0t〈(B+R)Jα(Yα(s)+ZA(s)),S*(t-s)h〉L2‍ds.
Moreover
(70)Jα(Yα(s)+ZA(s))⟶Y(s)+ZA(s),asα⟶0.
Notice that (B+R)(Jα(Yα(s)+ZA(s))→(B+R)(Yα(s)+ZA(s)) weakly converges in L2. So, letting α→0, we obtain
(71)〈Y(t),h〉L2=〈S(t)U0,h〉L2+∫0t‍〈(S(t-s)(B+R)(Y(s)+ZA(s)),h〉L2ds.
It follows from the arbitrariness of h that
(72)Y(t)=S(t)U0+∫0t‍S(t-s)(B+R)(Yα(s)+ZA(s))ds,t∈[0,T].
Thus, Y(t) is a mild solution of random Boussinesq equation (50).Theorem 18.
For anyU0∈H, the map φ:𝕋×Ω×H→H defined by the solution of stochastic Boussinesq equation (11) as U(t)=Φ(t,ϑt(ω))U0 has the cocycle property; that is, the solution of stochastic Boussinesq equation (11) generates a random dynamical system (Ω,ℱ,ℙ,(ϑt)t≥0,Φ).Proof.
From Theorem17, stochastic Boussinesq equation (11) admits a unique solution V(t,Z(ω)(t),x). Define the map
(73)Φ:ℝ+×Ω×H⟶H,Φ(t,ω)x=V(t·Z(ω)(t))(x-Z(ω)(0))+Z(ω)(t+s).(i)
By the similar argument of Theorem17, every solution Yα(t) of the Yosida approximation equation (50) is measurable. Notice that Yα(t)→Y(t) uniformly as α→0. Hence, the limit function Y(t) is also measurable. Thus, the mapping Φ is measurable.(ii)
Obviously,Φ(0,ω)=I.(iii)
It suffices to verify that the cocycle property holds for the mappingΦ, that is,
(74)Φ(t+s,ω)x=V(t+s,ZA(ω)(t+s))(x-ZA(ω)(0))+ZA(ω)(t+s).
In fact, recalling thatZA(ω)(s)=ZA(θsω)(0), it follows that
(75)Φ(t,θsω)[Φ(s,ω)x]=V(t,ZA(θsω)(t))(Φ(s,ω)x-ZA(θsω)(0))+ZA(θsω)(t)=V(t,ZA(θsω)(t))×[V(s,ZA(ω)(s))(x-ZA(ω)(0))+Z(ω)(s)-Z(θsω)(0)]+Z(θsω)(t)=V(t,ZA(θsω)(t))V(s,ZA(ω)(s))×(x-ZA(ω)(0))+ZA(θsω)(t)=V1(t).
Moreover,(76)V(t+s,ZA(ω)(t+s))(x-ZA(ω)(0))=V(t,ZA(θsω)(t))V(s,ZA(ω)(s))(x-ZA(ω)(0))=V2(t).
Since(77)V(0,ZA(θsω)(0))(x-ZA(θsω)(0))=x-ZA(θsω)(0).
Thus,(78)V1(0)=V(s,ZA(ω)(s))(x-ZA(ω)(0))=V(0,ZA(θsω)(0))V(s,ZA(ω)(s))×(x-ZA(ω)(0))=V2(0),dV1(t)dt=dV((t+s),ZA(ω))dt(t+s).
Therefore, we obtain(79)dV1(t)dt+AV1(t)+B(V1(t)+ZA(ω)(t+s),V1(t)+ZA(ω)(t+s))=-R(V1(t)+ZA(θt+sω)),dV2(t)dt+AV2(t)+B(V2(t)+ZA(θsω)(t),V2(t)+ZA(θsω)(t))=-R(V2(t)+ZA(θsω)(t)).
The uniqueness of the solution implies that almost surelyV1(t)=V2(t) holds, that is,
(80)Φ(t,θsω)[Φ(s,ω)x]=Φ(t+s,θt+s(ω))x.
Thus, the cocycle property for the mappingΦ holds.
By the definition of random dynamical systems [18], the solution mapping of the stochastic Boussinesq equation (11) generates a random dynamical system Φ. Thus, the proof of Theorem 18 is complete.


## 4. Discussion

In Section3, we have studied the long-time behavior of stochastic Boussinesq equations (1) driven by subordinator Lévy noise and have shown the cocycle property of random dynamical systems generated by the mild solution of stochastic Boussinesq equation (1). To prove the existence of random attractor, it suffices to show the existence of random absorbing set and the compactness of random dynamical system Φ, we refer the similar argument to [13].Here, we are also interested in the stochastic Boussinesq equations driven by Poisson noise and Wiener noise, and we are trying to show the existence of random dynamical systems. To the end, we consider the following stochastic Boussinesq equations driven by Lévy noises followed as(81)∂u∂t+(u·∇)u-νΔu+∇p=θe2+b1dt+dW1(t)+∫X‍f(x)N~1(dt,dx),∂θ∂t+(u·∇)θ-kΔθ=u2+b2dt+dW2(t)+∫X‍g(x)N~2(dt,dx),∇·u=0,u|∂D=0,u(0)=u0,θ(0)=θ0,
where W1(·) and W2(·) are H-valued Brownian motion, b1 and b2 are constants vector in H, f and g are measurable mappings from some measurable space X to H, and N~1 and N~2 are compensated Poisson measure on [0,∞)×X with intensity measure nν1 and nν2, respectively, where ν1 and ν2 are σ-finite measure on ℬ(X), f(x), and g(x)   satisfying
(82)∫U‍|f(x)|2eα|f(x)|ν(dx)<∞,∫U‍|g(x)|2eβ|g(x)|ν(dx)<∞,∀α>0,∀β>0.LetD([0,T],H) be the space of all cádlág paths from [0,T] to H endowed with the uniform convergence topology. Since there are finite jumps when the character measure λ(Z)<∞, we can rearrange the jump time of N(dt,dx) as σ1(ω)<σ2(ω)<⋯. Since there is no jump on the interval [0,σ1(ω)), just as the approach in [19], we can apply Banach fixed point theorem to prove that there exists a unique solution ϕ(t) in L2([0,σ1(ω));V)∩D([0,σ1(ω));H). Define
(83)ϕ(1)(t)={ϕ(t),0≤t<σ1(ω),ϕ(σ1-)+f(ϕ(σ1-),Pσ1),t=σ1(ω).
On [σ1(ω),σ2(ω)), define
(84)ϕ~0=ϕ(1)(σ1)1(σ1<∞),σ~2=(σ2-σ1)1(σ1<∞)+∞1(σ1=∞),ℱ~t=ℱσ1+t,P~t=(θσ1P)t1(σ1<∞).
Similar to the argument in [11], since Pt is stationary Poisson point process on R+×Z with intensity measure λ(dx)dt, then P~t is also a stable Poisson point process on R+×Z with intensity measure λ(dx)dt. Define
(85)ϕ(2)(t)={ϕ(1)(t),0≤t<σ1(ω),ϕ~(2)(t-σ1),σ1(ω)<t<σ2(ω),ϕ~(2)((σ2-σ1)-)+f(ϕ~(2)((σ2-σ1)-),Pσ2),t=σ2(ω),ϕ(n)(t)={ϕ(n-1)(t),t<σn-1(ω),ϕ~(n)(t-σn-1),σn-1(ω)<t<σn(ω),ϕ~(n)((σn-σn-1)-)+f(ϕ~(n)((σn-σn-1)-),Pσn),t=σn.
Hence, ϕ(n)(t) is cádlág on [0,T] such that B(ϕ(n),ϕ(n))∈H and Ap(ϕ(n)∈H, P a.s. for all t≥0, and
(86)P(∫0t‍[|ϕ(s)|+|B(ϕ(s)+zA(s),ϕ(s)+zA(s))|+2μ0|R(ϕ(s)+zA(s))|]ds<∞∫0t‍)=1,∀t>0.
Therefore, ϕ(n)(t) is a unique global weak solution of (81). We can verify the existence of random dynamical systems generated by the global weak solution of (81).



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