# Existence of Random Attractor for Stochastic Fractional Long-Short Wave Equations with Periodic Boundary Condition

Authors: Na Liu; Jie Xin
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

We consider the asymptotic behaviors of stochastic fractional long-short equations driven by a random force. Under a priori estimates in the sense of expectation, using Galerkin approximation by the stopping time and the Borel-Cantelli lemma, we prove the existence and uniqueness of solutions. Then a global random attractor and the existence of a stationary measure are obtained via the Birkhoff ergodic theorem and the Chebyshev inequality.


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## Body

## 1. Introduction

Long-short wave resonance equations arise in the study of the interaction of the surface waves with both gravity and capillary modes presence and also in the analysis of internal waves, as well as Rossby wave in [1]. In the plasma physics, they describe the resonance of the high-frequency electron plasma oscillation and associated low-frequency ion density perturbation in [2]. Long wave and short wave equations with periodic boundary condition have attracted considerable attention as a result of their rich physical and mathematical properties. Guo [3, 4] proved the existence of global solutions for long-short wave equations and generalized long-short wave equations. The existence of global attractor was studied in [5–9].The stochastic partial differential equation (SPDE) is a kind of partial differential equation with random term and random coefficients, which come from the random environmental effects and the errors of measurement. SPDE is used to describe better complex phenomenon, for example, quantum field theory, statistical mechanics, and financial mathematics; see [10–12] and so on. In [13–19], the authors obtained the existence and uniqueness of the solution and of attractors for SPDEs.In this paper, we consider that the random environmental effects and the errors of measurement are included into the model of fractional long-short wave equations. More specifically, we study the following equations:(1)iut--Δαu-nu+iδu=f+W˙1,(2)nt+βn+ux2=g+W˙2with the initial condition(3)ux,0=u0x,nx,0=n0x,and periodic boundary condition (4)ux,t=ux+2π,t,nx,t=nx+2π,t=0,where x∈D=[0,2π] in R1; t≥τ∈R+; δ,β,α>0. The unknown complex valued function u(x,t) is short surface wave packet and the unknown real valued function n(x,t) is a long interfacial wave. W1 and W2 are independent L2(D) value Wiener processes which are from the errors of measurement or the random environmental effects and can be seen in detail in the next section.Since the solutionu(x,t), if it exists, is a 2π-periodic function, we have the Fourier expansion (5)ux,t=∑k∈Zakteikx,where ak(t):R+→R,x∈R.Hence,(6)∂xu=∑k∈Zikakteikx,and (-▵)αu is defined by (7)-▵αu=∑k∈Zk2αakteikx.Since(8)eirx2=1,∫Deirxdx=0,∀r∈R,the following definitions make sense. Let(9)A=u∣u=∑k∈Zakteikx,∑k∈Zk4αak2t<∞,∑k∈Zakt2<∞,and let H2α be a complete distance space of the set A under the norm (10)uH2α=2π∑k∈Zk4αak2t1/2+2π∑k∈Zakt21/2.It is easy to get it to be a Banach space, and that ∀u∈H2α,u is space-periodic with the period 2π and its 2α order derivatives are in Lper2(D). And ∀u,v∈Hα, (11)ddtuv=uddtv+vddtu,-▵αu,v=-▵α1u,-▵α2v,when α1+α2=α,0≤α1,α2≤α. H2α is a Hilbert space with the inner product (12)u,vH2α=-▵αu,-▵αv.The rest of this paper is arranged as follows. In Section2, we present some preliminaries results. In Section 3, we give a series of time uniform a priori estimates in different energy spaces which will be used to prove our main results; see [20]. In Section 4, we show the existence and uniqueness of solutions for (1)-(2). In Section 5, the random weak attractor and the stationary measure are constructed.


## 2. Preliminary

In this paper, we outline the variational framework for studying problems (1)-(2) and list some preliminary lemmas which will be used later.We define a complete probability space(Ω,F,{Ft}t≥0,P). E denotes the expectation operator with respect to P. Stochastic terms W1(t) and W2(t) are defined on (Ω,F,{Ft}t≥0,P) by(13)W1t=q1xω1t,W2t=q2xω2t,where ω1(t) is a standard complex valued Wiener process, ω2(t) is a standard real valued Winer process independent of ω1(t), and q1(x),q2(x) are sufficiently smooth functions in a manner. The different inner product spaces for the solution (u,n) of (1)-(2) are defined as (14)V0=H0αD×L2D,V1=H2αD∩H0αD×H01D,V2=φ∈H3αD∩H0αD:-Δαφ∈H0αD×H2D∩H01D.Endow each Vi(i=0,1,2) with the usual norm, and satisfy V2⊂V1⊂V0 with compact embedding.Let(X,·X)⊂(Y,·Y)⊂(Z,·Z) be three Banach reflective spaces which satisfy X⊂Y with compact and dense embedding. The Banach space can be defined as follows: (15)G=v:v∈L20,T;X,dvdt∈L20,T;Zendowed with the natural norm (16)vG2=∫0TvXds+∫0Tdvdt2Zds,v∈G.We obtain the following lemma with regard to compactness result by [21].Lemma 1.
IfK is bounded in G, then K is precompact in L2(0,T;Y).In order to get maximal estimates on stochastic integrals, we need another lemma.U and H are Hilbert spaces which are separable and W is a Q-Wiener process on U0 with U0=Q1/2. Let L20=L20(U0,H) be the space of Hilbert-Schmidt operators from U0 to H. For such operators, we obtain the following lemma by [22].Lemma 2.
For anyr≥1 and any L20-valued predictable process Φ(t),t∈[0,T], we get (17)E∫0TΦsdWs=0,(18)Esup⁡∫0sΦσdWσ2r≤crsup⁡E∫0sΦσdWσ2r≤CrE∫0tΦsL202,where cr and Cr are some positive constants dependent on r.


## 3. Uniform a Priori Estimates in Time

### 3.1. A Priori Estimates inV0

Lemma 3.
Provided thatu0,f,q1∈L2(D), then for any T>0 and p≥1, we obtain u∈L2p(Ω;L∞([0,T];L2(D)))∩L∞([0,∞);L2p(Ω;L2(D))).Proof.
Taking the inner product of (1) with 2u, we get (19)iut--Δαu-nu+iδu,2u=f+W˙1,2u.Taking the imaginary part of (19) and applying the Itô formula to u2, one gets (20)ddtu2=-2δu2+2Im⁡∫Dfu-dx+2Im⁡∫Du-W˙1dx+q12.By (20), using Hölder’s and Young’s inequalities, we can obtain (21)ddtu2+δu2≤1δf2+q12+2Im⁡∫Du-W˙1dx.On the one hand, multiplying by eδt and integrating from 0 to t on both sides of (21), we get (22)Eu2≤e-δtEu02+1δ2f2+1δq12≤C,t>0,where C is independent of T.
On the other hand, integrating from0 to t and taking the supremum and the expectation on both sides of (21), we can obtain (23)Esup0≤t≤T⁡u2≤Eu02+1δf2+q12T+Esup0≤t≤T⁡∫0tIm⁡∫Du-W˙1dxds2+1.By Lemma 2, for any positive constant C, we obtain (24)Esup0≤t≤T⁡∫0tIm⁡∫Du-W˙1dxds2≤Cq12E∫0Tu2ds.From (24), for any T>0, there exists a positive constant CT such that (25)Esup0≤t≤T⁡u2≤Eu02+1δf2+q12T+Cq12E∫0Tu2ds+1≤CTEu04+f4+q14+1.
On the basis of the above estimates, we can further give an estimate ofu(t)02p for any p≥1. Applying the Itô formula and Hölder’s inequality, we obtain (26)ddtu2p≤-δp2u2p+cf2p+q12p+2pu2p-1Im⁡∫Du-W˙1dx.On the one hand, multiplying by e(δp/2)t, integrating from 0 to t, and taking the expectation on both sides of (26), we can have (27)Eu2p≤e-δp/2tEu02p+cf2p+q12p≤C,t>0,where C is independent of T.
On the other hand, by (26), we have (28)ddtu2p≤cf2p+q12p+2pu2p-1Im⁡∫Du-W˙1dx.Integrating from 0 to t and taking the supremum and the expectation on both sides of (28), we get (29)Esup0≤t≤T⁡u2p≤Eu02p+cf2p+q12pT+1+Esup0≤t≤T⁡∫0tpu2p-1Im⁡∫Du-W˙1dxds2.By Lemma 2, for any positive constant C, we have (30)Esup0≤t≤T⁡∫0tpu2p-1Im⁡∫Du-W˙1dxds2≤Cp2q12E∫0Tu4p-2ds,where 2p-1≥1. Then inserting (30) into (29), and by (27), for any T>0, we obtain (31)Esup0≤t≤T⁡u2p≤CTu0,f,q1,where CT(u0,f,q1) is a positive constant depending on u0,f,q1, and T.Lemma 4.
Provided that(u0,n0)∈V0,q1∈H0α(D),q2∈L2(D),f,g∈L2(D), then for any T>0 and p≥1, we have (u,n)∈L∞((0,∞);L2p(Ω;V0))∩L2p(Ω;L∞(0,T;V0)).Proof.
Taking the inner product of (2) with 2n, we have (32)nt+βn+ux2,2n=g+W˙2,2n.
Applying the Itô formula ton2, since nt=-ux2-βn+g+W˙2, we get (33)ddtn2-q22+2∫Dnux2dx+2βn2-2∫Dgndx-2∫DnW˙2dx=0.From (1), we get that (34)∫Dnux2dx=2Re⁡∫Du-xundx=-2Re⁡∫Du-x-iut+-Δαu-iδu+f-W˙1dx=∫Diδuxu¯-ux¯udx-2Re⁡∫Duf-xdx-2Re⁡∫DuW˙-1xdx.Notice that (35)ddt∫Diuxu--uu-xdx=2∫Diuxu-t-utu-xdx+2Im⁡∫Dq1q-1xdx.From (33), (34), and (35), we can have (36)ddtn2+∫Diuxu--uu-xdx+2βn2=-2∫Diδuxu--u-xudx+4Re⁡∫Duf-xdx+2Im⁡∫Dq1q-1xdx+q22+4Re⁡∫DuW˙-1xdx+2∫Dgndx+2∫DnW˙2dx.Applying the Itô formula to ux2, and taking the inner product of (1) with (ut+δu), we obtain (37)iut--Δαu-nu+iδu,ut+δu=2f+W˙1,ut+δu.Taking the real part of (37), we get (38)ddt-Δα/2u2+2δ-Δα/2u2+2Re⁡∫Dnuu-tdx+2δ∫Dnu2dx+2Re⁡∫Du-t+δu-W˙1dx+2Re⁡∫Dfu-tdx+2δRe⁡∫Dfu-dx=q1x2.It is easy to check that (39)ddt∫Dnu2dx=∫Dntu2dx+2Re⁡∫Dnutu-dx+2∫Dq2Im⁡q1u-dx.Otherwise, by (2), we can have nt=-ux2-βn+g+W˙2.
So(40)∫Dntu2dx=∫D-ux2-βn+g+W˙2u2dx=-β∫Dnu2dx+∫Dgu2dx+∫DW˙2u2dx,-2Re⁡∫Du-t+δu-W˙1dx=2Im⁡∫Diu-t+iδu-W˙1dx=2Im⁡∫Df¯+W˙1¯+-Δαu¯+nu¯W˙1dx=2Im⁡∫D-Δα/2u-Δα/2W˙1dx+2Im⁡∫Dnu-+f-W˙1dx.From (2), we can obtain (41)ddt-Δα/2u2+2Re⁡∫Dfu-dx+∫Dnu2dx+2δ-Δα/2u2=-2δ∫Dnu2dx-2δRe⁡∫Dfu-dx+q1x2+2Im⁡∫D-Δα/2u-Δα/2W˙1dx+2Im⁡∫Dnu-+f-W˙1dx+2∫Dq2Im⁡q1u-dx-β∫Dnu2dx+∫Dgu2dx+∫DW˙2u2dx.Using Hölder’s inequality, the Gagliardo-Nirenberg inequality, and Young’s inequality, we can estimate each term. Now, let H0=-Δα/2u2+2Re⁡∫Dfu-dx+∫Dnu2dx+n2+∫Di(uxu--uu-x)dx and take η=min⁡{δ,β}. Putting (36) and (41) into the result, we get (42)ddtH0+ηH0≤cf,g,q1,q2+cu6+∫Du2W˙2dx+2∫DnW˙2dx-2Im⁡∫D-Δα/2u-Δα/2W˙1dx-2Im⁡∫Dnu-+f-W˙1dx+4Re⁡∫DuW˙-1xdx.Multiplying by eηt, integrating from 0 to t, and taking expectation on both sides of (42) yield (43)EH0≤e-ηtEH00+cf,g,q1,q2+cE∫0te-ηt-su6ds.From (27), we can estimate (43) and obtain (44)EH0t≤e-βtEH00+cf,g,q1,q2,u0≤cf,g,q1,q2,u0≤C,t>0,where C is independent of T.
Since(45)H0t≥12-Δα/2u2+n2-cu2+g2+u6,for any t>0, we obtain (46)E-Δα/2u2+n2≤cEu2+g2+u6+cEH0t≤C,where C is independent of T.
Besides, we estimateH0p(t) for p≥1. Firstly, applying the Itô formula to H0p(t), we have (47)ddtH0pt≤-ηp2H0pt+cu6p+c+pH0p-1t∫Du2W˙2dx+2∫DnW˙2dx+pH0p-1t-2Im⁡∫D-Δα/2u-Δα/2W˙1dx-2Im⁡∫nu-+f-W˙1dx+4Re⁡∫DuW˙-1xdx.Multiplying by e(ηp/2)t, integrating from 0 to t, and taking expectation on both sides of (47) yield (48)EH0pt≤e-ηp/2tEH0p0+c+cE∫0te-ηp/2t-su6pds.From (27) and (48), we obtain (49)EH0pt≤eηp/2tEH0p0+c≤C,t>0,where C is independent of T.
Consequently, from (45), we obtain (50)E-Δα/2u2+n2≤C,t>0.On the one side, integrating from 0 to t on both sides of (42), we can deduce (51)H0t≤H00+cf,g,q1,q2t+c∫0tu6ds+∫0t∫Du2W˙2dx+2∫DnW˙2dxds+∫0t-2Im⁡∫D-Δα/2u-Δα/2W˙1dx-2Im⁡∫Dnu-+f-W˙1dx+4Re⁡∫DuW˙-1xdxds.Taking the supremum and expectation on both sides of (51) yields (52)Esup0≤t≤T⁡H0t≤EH00+cf,g,q1,q2T+cEsup0≤t≤T⁡∫0tu6ds+Esup0≤t≤T⁡∫0t∫Du2W˙2dxds2+Esup0≤t≤T⁡∫0t∫DnW˙2dxds2+Esup0≤t≤T⁡∫0tIm⁡∫DIm⁡∫D-Δα/2u-Δα/2W˙1dxds2+Esup0≤t≤T⁡∫0tIm⁡∫Dnu-+f-W˙1dxds2+Esup0≤t≤T⁡∫0tRe⁡∫DuW˙1xdxds2.And then by estimating each term of the right hand side of (52), we obtain (53)Esup0≤t≤TH0t≤CTE0,f,g,q1,q2.On the other side, for H0p(t)(p≥1), integrating from 0 to t and taking the supremum and the expectation on both sides of (47) yield (54)Esup0≤t≤T⁡H0pt≤EH0p0+cq1,q2T+cEsup0≤t≤T⁡∫0tu6pds+c+p2Esup0≤t≤T⁡∫0tH0p-1s∫u2W˙2dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1s∫nW˙2dxds2+p2Esup0≤t≤T⁡∫0tH0p-1s4Re⁡∫D-Δα/2u-Δα/2W˙1dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫DuW˙-1xdxds2+p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫nu-+f-W˙1dxds2.Now we estimate each term of (54). For the third term on the right hand of (54), (55)cEsup0≤t≤T⁡∫0tu6pds≤cE∫0Tu6pds≤CT.For the fifth and sixth terms on the right hand of (54), we have (56)p2Esup0≤t≤T⁡∫0tH0p-1s∫u2W˙2dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1s∫nW˙2dxds2≤cq22E∫0TH02ps+u2p+u6p+f2pds≤CT.For the seventh and eight terms on the right hand of (54), using a similar method, we estimate (57)p2Esup0≤t≤T⁡∫0tH0p-1s4Re⁡∫D-Δα/2u-Δα/2W˙1dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫DuW˙-1xdxds2≤c-Δα/2q12E∫0TH02ps+u2p+u6p+f2pds≤CT.For the last term on the right hand of (54), we estimate (58)p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫nu-+f-W˙1dxds2≤c-Δα/2q12E∫0TH02ps+u2p+u6p+f2pds≤CT.By the above estimates, we clearly obtain (59)Esup0≤t≤T⁡H0pt≤EH0p0+CTq1,q2,E0,n1,n0<CTq1,q2,E0,n1,n0.In addition, by (46), we can get (60)Esup0≤t≤T⁡-Δα/2u2p+n2p≤cEsup0≤t≤T⁡H0pt+cEsup0≤t≤T⁡u2p+u6p+f2p≤CT.Then we obtain (61)Esup0≤t≤T⁡u,nV02p≤CTf,g,q1,q2,u0,n0.Then the proof is complete.


### 3.2. A Priori Estimates inV1

Lemma 5.
Provided thatf,g∈H1(D),(u0,n0)∈V1,q1∈H2α,q2∈H1(D), then for any T>0 and p≥1, we have (n,u)∈L∞((0,∞);L2p(Ω;V1))∩L2p(Ω;L∞(0,T;V1)).Proof.
Taking the inner product of (1) with (-Δ)αut, we get (62)iut--Δαu-nu+iδu,-Δαut=f+W˙1,-Δαut.Applying the Itô formula to (-Δ)αu2, we have (63)ddt-Δαu2=2Re⁡∫D-Δαu-Δαu-tdx+-Δαq12.Since ut=i(-(-Δ)αu-nu+iδu-f-W˙1) and (64)ddtRe⁡∫nu-Δαu-dx=Re⁡∫Dntu-Δαu-+nut-Δαu+nu-Δαu-t+q2-Δαq-1+iq2-Δαq-1u-q2q1-Δαu-dx,we obtain (65)ddt-Δαu2-2Re⁡∫Dnu-Δαu-dx+2Re⁡∫Df-Δαu-dx+2α-Δαu2=-Δαq12+2q1x2+2Re⁡∫Diq2q1-Δαu--q2-Δαq-1udx-2Re⁡∫Dnut-Δαudx+2δRe⁡∫Dnu-Δαu-dx-2δRe⁡∫Df-Δαudx-2Re⁡∫DW˙1-Δαu-t+δ-Δαu-dx-2Re⁡∫Dntu-Δαu-dx.
According to the above inequality, we can estimate each term on the right hand side of (65) using Höder’s inequality, the Gagliardo-Nirenberg, and Young’s inequality. So, by (65), we get (66)ddt-Δαu2-2Re⁡∫Dnu-Δαu-dx+2Re⁡∫Df-Δαu-dx+2α-Δαu2≤δ2-Δαu2+β2nx2+cq2H14+q1H2α2+g4+f8+uH18+n12-2Re⁡∫DW˙1-Δαu-ty-2Re⁡∫Du-Δαu-W˙2dx+2Re⁡∫Din-Δαu-W˙1dx.Otherwise, applying the Itô formula to nx2, we have (67)ddtnx2=2∫Dnxnxtdx+q2x2.Since nt=g+W˙2-βn-ux2 and (68)ddti∫Dux-Δαu---Δαuu¯xdx=2i∫Duxt-Δαu--u-xt-Δαudx+2iRe⁡∫Dq1x-Δαq-2dx,we obtain (69)ddtnx2+i∫Dux-Δαu---Δαuu¯xdx+2βnx2≤δ2-Δαu2+β2nx2+cgx2+fx2+q1H2α2+q2x2+2∫DnxW˙2xdx+4Re⁡∫D-Δαu-W˙1xdx.Thus, by (66) and (69), taking η=min⁡{δ,β}, and letting (70)H1t=-Δαu2+nx2-2Re⁡∫Dnu-Δαu-dx+2Re⁡∫Df-Δαu-dx+i∫Dux-Δαu---Δαuu¯xdx,we get (71)ddtH1t+ηH1t≤cuH18+n12+q2H14+q1H2α4+gH14+fH18+1-2Re⁡∫DW˙1-Δαu-ty-2Re⁡∫Du-Δαu-W˙2dx+2Re⁡∫Din-Δαu-W˙1dx+2∫DnxW˙2xdx+4Re⁡∫D-Δαu-W˙1xdx.Multiplying by eηt, integrating from 0 to t, and taking expecting on both sides of (71), by (46), we obtain (72)EH1t≤e-βtEH10+cf,g,q1,q2,u0≤cf,g,q1,q2,u0≤C,t>0,where C is independent of T. Since (73)H1t≥12-Δαu2+nx2-cux8+n4+1+f2,for any t>0, we obtain (74)E-Δαu2+nx2≤cEux8+n4+1+f2+cEH1t≤C.where C is independent of T.
Next step, we estimateH1p(t) for p≥1. On the one hand, applying the Itô formula to H1p(t), we obtain(75)EH1pt≤e-ηt/2tEH1p0+c+cE∫0te-ηt/2t-s-Δαu8p+n12p+1ds.By (45), we obtain (76)EH0pt≤e-ηp/2tEH0p0+c≤C,t>0,where C is independent of T. Thus, from (73), for any t>0, we get (77)E-Δαu2p+nx2p≤C.
Integrating from0 to t and taking the supremum and expectation on both sides of (71), as with the estimates in Lemma 4 for each term, we deduce (78)Esup0≤t≤T⁡H1t≤CTf,g,q1,q2,u0.On the other hand, for H0p(t)(p≥1), integrating from 0 to t and taking the supremum and the expectation on both sides of (71), and estimating each term, we get (79)Esup0≤t≤T⁡H1pt≤EH1p0+CTf,g,q1,q2,u0,n0≤CTf,g,q1,q2,u0,n0.Therefore, from (73), it is inferred that for p≥1(80)Esup0≤t≤T⁡-Δαu2p+nx2p≤CTf,g,q1,q2,u0,n0.So then, we have (81)Esup0≤t≤T⁡u,nV12p≤CTf,g,q1,q2,u0,n0,and the proof is complete.


### 3.3. A Priori Estimates inV2

By using the similar method and idea as Sections3.1 and 3.2, we can achieve a priori estimates in V2. For simplicity, we only provide the idea of the proof. Using (1)-(2) and applying the Itô formula to (-Δ)αn2 and (-Δ)αux2, respectively, we can get some inequalities by Höder’s inequality, the Gagliardo-Nirenberg, and Young’s inequality. After that, taking the supremum and expectation for inequalities and estimating on (-Δ)αn2 and (-Δ)αux2 by Gronwall-type, we can deduce the following lemma.Lemma 6.
Provided that(u0,n0)∈V2,q2∈H2(D),q1∈H3α(D), and f,g∈H2(D), then for any T>0 and p≥1, we have (u,n)∈L∞(0,∞;L2p(Ω;V2))∩L2p(Ω;L∞(0,T;V2)).


## 3.1. A Priori Estimates inV0

Lemma 3.
Provided thatu0,f,q1∈L2(D), then for any T>0 and p≥1, we obtain u∈L2p(Ω;L∞([0,T];L2(D)))∩L∞([0,∞);L2p(Ω;L2(D))).Proof.
Taking the inner product of (1) with 2u, we get (19)iut--Δαu-nu+iδu,2u=f+W˙1,2u.Taking the imaginary part of (19) and applying the Itô formula to u2, one gets (20)ddtu2=-2δu2+2Im⁡∫Dfu-dx+2Im⁡∫Du-W˙1dx+q12.By (20), using Hölder’s and Young’s inequalities, we can obtain (21)ddtu2+δu2≤1δf2+q12+2Im⁡∫Du-W˙1dx.On the one hand, multiplying by eδt and integrating from 0 to t on both sides of (21), we get (22)Eu2≤e-δtEu02+1δ2f2+1δq12≤C,t>0,where C is independent of T.
On the other hand, integrating from0 to t and taking the supremum and the expectation on both sides of (21), we can obtain (23)Esup0≤t≤T⁡u2≤Eu02+1δf2+q12T+Esup0≤t≤T⁡∫0tIm⁡∫Du-W˙1dxds2+1.By Lemma 2, for any positive constant C, we obtain (24)Esup0≤t≤T⁡∫0tIm⁡∫Du-W˙1dxds2≤Cq12E∫0Tu2ds.From (24), for any T>0, there exists a positive constant CT such that (25)Esup0≤t≤T⁡u2≤Eu02+1δf2+q12T+Cq12E∫0Tu2ds+1≤CTEu04+f4+q14+1.
On the basis of the above estimates, we can further give an estimate ofu(t)02p for any p≥1. Applying the Itô formula and Hölder’s inequality, we obtain (26)ddtu2p≤-δp2u2p+cf2p+q12p+2pu2p-1Im⁡∫Du-W˙1dx.On the one hand, multiplying by e(δp/2)t, integrating from 0 to t, and taking the expectation on both sides of (26), we can have (27)Eu2p≤e-δp/2tEu02p+cf2p+q12p≤C,t>0,where C is independent of T.
On the other hand, by (26), we have (28)ddtu2p≤cf2p+q12p+2pu2p-1Im⁡∫Du-W˙1dx.Integrating from 0 to t and taking the supremum and the expectation on both sides of (28), we get (29)Esup0≤t≤T⁡u2p≤Eu02p+cf2p+q12pT+1+Esup0≤t≤T⁡∫0tpu2p-1Im⁡∫Du-W˙1dxds2.By Lemma 2, for any positive constant C, we have (30)Esup0≤t≤T⁡∫0tpu2p-1Im⁡∫Du-W˙1dxds2≤Cp2q12E∫0Tu4p-2ds,where 2p-1≥1. Then inserting (30) into (29), and by (27), for any T>0, we obtain (31)Esup0≤t≤T⁡u2p≤CTu0,f,q1,where CT(u0,f,q1) is a positive constant depending on u0,f,q1, and T.Lemma 4.
Provided that(u0,n0)∈V0,q1∈H0α(D),q2∈L2(D),f,g∈L2(D), then for any T>0 and p≥1, we have (u,n)∈L∞((0,∞);L2p(Ω;V0))∩L2p(Ω;L∞(0,T;V0)).Proof.
Taking the inner product of (2) with 2n, we have (32)nt+βn+ux2,2n=g+W˙2,2n.
Applying the Itô formula ton2, since nt=-ux2-βn+g+W˙2, we get (33)ddtn2-q22+2∫Dnux2dx+2βn2-2∫Dgndx-2∫DnW˙2dx=0.From (1), we get that (34)∫Dnux2dx=2Re⁡∫Du-xundx=-2Re⁡∫Du-x-iut+-Δαu-iδu+f-W˙1dx=∫Diδuxu¯-ux¯udx-2Re⁡∫Duf-xdx-2Re⁡∫DuW˙-1xdx.Notice that (35)ddt∫Diuxu--uu-xdx=2∫Diuxu-t-utu-xdx+2Im⁡∫Dq1q-1xdx.From (33), (34), and (35), we can have (36)ddtn2+∫Diuxu--uu-xdx+2βn2=-2∫Diδuxu--u-xudx+4Re⁡∫Duf-xdx+2Im⁡∫Dq1q-1xdx+q22+4Re⁡∫DuW˙-1xdx+2∫Dgndx+2∫DnW˙2dx.Applying the Itô formula to ux2, and taking the inner product of (1) with (ut+δu), we obtain (37)iut--Δαu-nu+iδu,ut+δu=2f+W˙1,ut+δu.Taking the real part of (37), we get (38)ddt-Δα/2u2+2δ-Δα/2u2+2Re⁡∫Dnuu-tdx+2δ∫Dnu2dx+2Re⁡∫Du-t+δu-W˙1dx+2Re⁡∫Dfu-tdx+2δRe⁡∫Dfu-dx=q1x2.It is easy to check that (39)ddt∫Dnu2dx=∫Dntu2dx+2Re⁡∫Dnutu-dx+2∫Dq2Im⁡q1u-dx.Otherwise, by (2), we can have nt=-ux2-βn+g+W˙2.
So(40)∫Dntu2dx=∫D-ux2-βn+g+W˙2u2dx=-β∫Dnu2dx+∫Dgu2dx+∫DW˙2u2dx,-2Re⁡∫Du-t+δu-W˙1dx=2Im⁡∫Diu-t+iδu-W˙1dx=2Im⁡∫Df¯+W˙1¯+-Δαu¯+nu¯W˙1dx=2Im⁡∫D-Δα/2u-Δα/2W˙1dx+2Im⁡∫Dnu-+f-W˙1dx.From (2), we can obtain (41)ddt-Δα/2u2+2Re⁡∫Dfu-dx+∫Dnu2dx+2δ-Δα/2u2=-2δ∫Dnu2dx-2δRe⁡∫Dfu-dx+q1x2+2Im⁡∫D-Δα/2u-Δα/2W˙1dx+2Im⁡∫Dnu-+f-W˙1dx+2∫Dq2Im⁡q1u-dx-β∫Dnu2dx+∫Dgu2dx+∫DW˙2u2dx.Using Hölder’s inequality, the Gagliardo-Nirenberg inequality, and Young’s inequality, we can estimate each term. Now, let H0=-Δα/2u2+2Re⁡∫Dfu-dx+∫Dnu2dx+n2+∫Di(uxu--uu-x)dx and take η=min⁡{δ,β}. Putting (36) and (41) into the result, we get (42)ddtH0+ηH0≤cf,g,q1,q2+cu6+∫Du2W˙2dx+2∫DnW˙2dx-2Im⁡∫D-Δα/2u-Δα/2W˙1dx-2Im⁡∫Dnu-+f-W˙1dx+4Re⁡∫DuW˙-1xdx.Multiplying by eηt, integrating from 0 to t, and taking expectation on both sides of (42) yield (43)EH0≤e-ηtEH00+cf,g,q1,q2+cE∫0te-ηt-su6ds.From (27), we can estimate (43) and obtain (44)EH0t≤e-βtEH00+cf,g,q1,q2,u0≤cf,g,q1,q2,u0≤C,t>0,where C is independent of T.
Since(45)H0t≥12-Δα/2u2+n2-cu2+g2+u6,for any t>0, we obtain (46)E-Δα/2u2+n2≤cEu2+g2+u6+cEH0t≤C,where C is independent of T.
Besides, we estimateH0p(t) for p≥1. Firstly, applying the Itô formula to H0p(t), we have (47)ddtH0pt≤-ηp2H0pt+cu6p+c+pH0p-1t∫Du2W˙2dx+2∫DnW˙2dx+pH0p-1t-2Im⁡∫D-Δα/2u-Δα/2W˙1dx-2Im⁡∫nu-+f-W˙1dx+4Re⁡∫DuW˙-1xdx.Multiplying by e(ηp/2)t, integrating from 0 to t, and taking expectation on both sides of (47) yield (48)EH0pt≤e-ηp/2tEH0p0+c+cE∫0te-ηp/2t-su6pds.From (27) and (48), we obtain (49)EH0pt≤eηp/2tEH0p0+c≤C,t>0,where C is independent of T.
Consequently, from (45), we obtain (50)E-Δα/2u2+n2≤C,t>0.On the one side, integrating from 0 to t on both sides of (42), we can deduce (51)H0t≤H00+cf,g,q1,q2t+c∫0tu6ds+∫0t∫Du2W˙2dx+2∫DnW˙2dxds+∫0t-2Im⁡∫D-Δα/2u-Δα/2W˙1dx-2Im⁡∫Dnu-+f-W˙1dx+4Re⁡∫DuW˙-1xdxds.Taking the supremum and expectation on both sides of (51) yields (52)Esup0≤t≤T⁡H0t≤EH00+cf,g,q1,q2T+cEsup0≤t≤T⁡∫0tu6ds+Esup0≤t≤T⁡∫0t∫Du2W˙2dxds2+Esup0≤t≤T⁡∫0t∫DnW˙2dxds2+Esup0≤t≤T⁡∫0tIm⁡∫DIm⁡∫D-Δα/2u-Δα/2W˙1dxds2+Esup0≤t≤T⁡∫0tIm⁡∫Dnu-+f-W˙1dxds2+Esup0≤t≤T⁡∫0tRe⁡∫DuW˙1xdxds2.And then by estimating each term of the right hand side of (52), we obtain (53)Esup0≤t≤TH0t≤CTE0,f,g,q1,q2.On the other side, for H0p(t)(p≥1), integrating from 0 to t and taking the supremum and the expectation on both sides of (47) yield (54)Esup0≤t≤T⁡H0pt≤EH0p0+cq1,q2T+cEsup0≤t≤T⁡∫0tu6pds+c+p2Esup0≤t≤T⁡∫0tH0p-1s∫u2W˙2dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1s∫nW˙2dxds2+p2Esup0≤t≤T⁡∫0tH0p-1s4Re⁡∫D-Δα/2u-Δα/2W˙1dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫DuW˙-1xdxds2+p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫nu-+f-W˙1dxds2.Now we estimate each term of (54). For the third term on the right hand of (54), (55)cEsup0≤t≤T⁡∫0tu6pds≤cE∫0Tu6pds≤CT.For the fifth and sixth terms on the right hand of (54), we have (56)p2Esup0≤t≤T⁡∫0tH0p-1s∫u2W˙2dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1s∫nW˙2dxds2≤cq22E∫0TH02ps+u2p+u6p+f2pds≤CT.For the seventh and eight terms on the right hand of (54), using a similar method, we estimate (57)p2Esup0≤t≤T⁡∫0tH0p-1s4Re⁡∫D-Δα/2u-Δα/2W˙1dxds2+p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫DuW˙-1xdxds2≤c-Δα/2q12E∫0TH02ps+u2p+u6p+f2pds≤CT.For the last term on the right hand of (54), we estimate (58)p2Esup0≤t≤T⁡∫0t2H0p-1sIm⁡∫nu-+f-W˙1dxds2≤c-Δα/2q12E∫0TH02ps+u2p+u6p+f2pds≤CT.By the above estimates, we clearly obtain (59)Esup0≤t≤T⁡H0pt≤EH0p0+CTq1,q2,E0,n1,n0<CTq1,q2,E0,n1,n0.In addition, by (46), we can get (60)Esup0≤t≤T⁡-Δα/2u2p+n2p≤cEsup0≤t≤T⁡H0pt+cEsup0≤t≤T⁡u2p+u6p+f2p≤CT.Then we obtain (61)Esup0≤t≤T⁡u,nV02p≤CTf,g,q1,q2,u0,n0.Then the proof is complete.


## 3.2. A Priori Estimates inV1

Lemma 5.
Provided thatf,g∈H1(D),(u0,n0)∈V1,q1∈H2α,q2∈H1(D), then for any T>0 and p≥1, we have (n,u)∈L∞((0,∞);L2p(Ω;V1))∩L2p(Ω;L∞(0,T;V1)).Proof.
Taking the inner product of (1) with (-Δ)αut, we get (62)iut--Δαu-nu+iδu,-Δαut=f+W˙1,-Δαut.Applying the Itô formula to (-Δ)αu2, we have (63)ddt-Δαu2=2Re⁡∫D-Δαu-Δαu-tdx+-Δαq12.Since ut=i(-(-Δ)αu-nu+iδu-f-W˙1) and (64)ddtRe⁡∫nu-Δαu-dx=Re⁡∫Dntu-Δαu-+nut-Δαu+nu-Δαu-t+q2-Δαq-1+iq2-Δαq-1u-q2q1-Δαu-dx,we obtain (65)ddt-Δαu2-2Re⁡∫Dnu-Δαu-dx+2Re⁡∫Df-Δαu-dx+2α-Δαu2=-Δαq12+2q1x2+2Re⁡∫Diq2q1-Δαu--q2-Δαq-1udx-2Re⁡∫Dnut-Δαudx+2δRe⁡∫Dnu-Δαu-dx-2δRe⁡∫Df-Δαudx-2Re⁡∫DW˙1-Δαu-t+δ-Δαu-dx-2Re⁡∫Dntu-Δαu-dx.
According to the above inequality, we can estimate each term on the right hand side of (65) using Höder’s inequality, the Gagliardo-Nirenberg, and Young’s inequality. So, by (65), we get (66)ddt-Δαu2-2Re⁡∫Dnu-Δαu-dx+2Re⁡∫Df-Δαu-dx+2α-Δαu2≤δ2-Δαu2+β2nx2+cq2H14+q1H2α2+g4+f8+uH18+n12-2Re⁡∫DW˙1-Δαu-ty-2Re⁡∫Du-Δαu-W˙2dx+2Re⁡∫Din-Δαu-W˙1dx.Otherwise, applying the Itô formula to nx2, we have (67)ddtnx2=2∫Dnxnxtdx+q2x2.Since nt=g+W˙2-βn-ux2 and (68)ddti∫Dux-Δαu---Δαuu¯xdx=2i∫Duxt-Δαu--u-xt-Δαudx+2iRe⁡∫Dq1x-Δαq-2dx,we obtain (69)ddtnx2+i∫Dux-Δαu---Δαuu¯xdx+2βnx2≤δ2-Δαu2+β2nx2+cgx2+fx2+q1H2α2+q2x2+2∫DnxW˙2xdx+4Re⁡∫D-Δαu-W˙1xdx.Thus, by (66) and (69), taking η=min⁡{δ,β}, and letting (70)H1t=-Δαu2+nx2-2Re⁡∫Dnu-Δαu-dx+2Re⁡∫Df-Δαu-dx+i∫Dux-Δαu---Δαuu¯xdx,we get (71)ddtH1t+ηH1t≤cuH18+n12+q2H14+q1H2α4+gH14+fH18+1-2Re⁡∫DW˙1-Δαu-ty-2Re⁡∫Du-Δαu-W˙2dx+2Re⁡∫Din-Δαu-W˙1dx+2∫DnxW˙2xdx+4Re⁡∫D-Δαu-W˙1xdx.Multiplying by eηt, integrating from 0 to t, and taking expecting on both sides of (71), by (46), we obtain (72)EH1t≤e-βtEH10+cf,g,q1,q2,u0≤cf,g,q1,q2,u0≤C,t>0,where C is independent of T. Since (73)H1t≥12-Δαu2+nx2-cux8+n4+1+f2,for any t>0, we obtain (74)E-Δαu2+nx2≤cEux8+n4+1+f2+cEH1t≤C.where C is independent of T.
Next step, we estimateH1p(t) for p≥1. On the one hand, applying the Itô formula to H1p(t), we obtain(75)EH1pt≤e-ηt/2tEH1p0+c+cE∫0te-ηt/2t-s-Δαu8p+n12p+1ds.By (45), we obtain (76)EH0pt≤e-ηp/2tEH0p0+c≤C,t>0,where C is independent of T. Thus, from (73), for any t>0, we get (77)E-Δαu2p+nx2p≤C.
Integrating from0 to t and taking the supremum and expectation on both sides of (71), as with the estimates in Lemma 4 for each term, we deduce (78)Esup0≤t≤T⁡H1t≤CTf,g,q1,q2,u0.On the other hand, for H0p(t)(p≥1), integrating from 0 to t and taking the supremum and the expectation on both sides of (71), and estimating each term, we get (79)Esup0≤t≤T⁡H1pt≤EH1p0+CTf,g,q1,q2,u0,n0≤CTf,g,q1,q2,u0,n0.Therefore, from (73), it is inferred that for p≥1(80)Esup0≤t≤T⁡-Δαu2p+nx2p≤CTf,g,q1,q2,u0,n0.So then, we have (81)Esup0≤t≤T⁡u,nV12p≤CTf,g,q1,q2,u0,n0,and the proof is complete.


## 3.3. A Priori Estimates inV2

By using the similar method and idea as Sections3.1 and 3.2, we can achieve a priori estimates in V2. For simplicity, we only provide the idea of the proof. Using (1)-(2) and applying the Itô formula to (-Δ)αn2 and (-Δ)αux2, respectively, we can get some inequalities by Höder’s inequality, the Gagliardo-Nirenberg, and Young’s inequality. After that, taking the supremum and expectation for inequalities and estimating on (-Δ)αn2 and (-Δ)αux2 by Gronwall-type, we can deduce the following lemma.Lemma 6.
Provided that(u0,n0)∈V2,q2∈H2(D),q1∈H3α(D), and f,g∈H2(D), then for any T>0 and p≥1, we have (u,n)∈L∞(0,∞;L2p(Ω;V2))∩L2p(Ω;L∞(0,T;V2)).


## 4. Proofs of Theorems7 and 8

Based on the prior estimate, we acquire the existence and uniqueness of a solution for the stochastic fractional long-short wave equations (1)-(2) in spaces V1.Theorem 7.
If(u0,n0)∈V1,q2∈H1(D),q1∈H2α(D), and f,g∈H1(D), then there exists a unique solution (u,n)∈(R+;V1), almost surely satisfying (1)-(2). In addition, (u,n) is continuous from R+ to V1.Proof.
First, we know that(n0,u0)∈V1. Let {ei(x)}i=1∞ be an orthonormal basis of eigenvectors of the Laplace operator on D, which is an orthonormal basis of L2(D). Consider Pk as the projection from L2(D) onto the space spanned by {ei:i=1,2,…,k}. Then the approximation solution (uk,nk) solves the approximation problem (82)iutk--Δαuk-Pknkuk+iδuk=W˙1k+fk,ntk+Pkuk2x+βnk=gk+W˙2k,where Pk is the projector onto the first k vectors ei,W˙1k=PkW˙1,W˙2k=PkW˙2, and Pk commutes with the operator (-Δ)α. We will treat the above equations pathwise by introducing the following random processes solving (83)iξtk--Δαξk+iδξk=W˙1k,ζk+βζk=W˙2k,with periodic boundary conditions and initial conditions (84)ζx,t=ηx+2π,t,ξx,t=ξx+2π,t=0,ζtx,0=0,ζx,0=0,ξ0,t=0,x∈D,whereδ>0,β>0.
In accordance with the same method as in Section3, for any T>0 and almost all ω∈Ω, we have (85)ζ∈C0,T;H01D,ξ∈C0,T;H2αD∩H0αD.Therefore, we can get the following estimate: (86)Eηx2+-Δαξ2≤C,where a positive constant C is independent of T. Moreover, for any T>0, (87)Esup0≤t≤T⁡ηx2+-Δαξ2≤CTholds for a positive constant CT dependent on T. Set Bk,M=(Uk,M,Nk,M) as the solution of the following equations: (88)iUtk,M--ΔαUk,M-χMBk,MV1Pknk,Muk,M+iδUk,M=fk,(89)Ntk,M+χMBk,MV1Pkuk,M2x+βNk,M=gk,(90)Nk,Mx,0=Pkn0,Uk,Mx,0=Pku0.Let nk,M=Nk,M+Pkζ,uk,M=Uk,M+Pkξ, and χM∈C0∞(R) satisfy χM(r)=1 for r≤M and χM(r)=0 for r≥2M. It is easy to see that (88)-(89) are random differential equations with Lipschitz nonlinearity in finite dimension. Afterwards, for almost all ω∈Ω, we have a unique solution (Uk,M,Nk,M) for (88)-(89). We define the stopping time as follows: (91)τM=inf⁡t>0:BV1k,M≥M,if the set {BMk,M≥M} is nonempty, or else τM=∞. Because τM is increasing in M, let τ∞=limM→∞⁡τM almost surely. For t<τM, we obtain (92)Uk,M,Nk,M+Pkξ,Pkζ,satisfying (82). On the basis of the estimates given in Section 3.2 and (86)-(87), for any t≥0, we obtain (93)EUk,M,Nk,MV12≤Cf,g,q1,q2,n0,u0,where the positive constant C(f,g,q1,q2,n0,u0), which is dependent on T and M. And for T>0, we obtain (94)Esup0≤t≤T∧τM⁡Uk,M,Nk,MV12≤CTf,g,q1,q2,n0,u0,with the positive constant C(f,g,q1,q2,n0,u0), which is dependent on T but independent of M. Let T∧τM=min⁡{T,τM}. On the other side, we get (95)EUk,MT∧τM,Nk,MT∧τMV12≥EIτM≤TUk,MT∧τM,Nk,MT∧τMV12≥M2Pτ≤T,where I(τM≤T)=1 for τM≤T and I(τM≤T)=0 for τM>T. Then, by (93), we get (96)PτM≤T≤1M2Cf,g,q1,q2,n0,u0.In the light of the above estimate and the Borel-Cantelli lemma, for any T>0, we obtain (97)Pτ∞>T=1.So we know that (98)Uk,Nk=limM→∞⁡Uk,M,Nk,Msatisfies the following random differential equations: (99)iUtk--ΔαUk-Pknk,uk+iδUk=fk,(100)Ntk+Pkuk,M2x+βNk=gk,with initial conditions (101)Uk0=Pku0,Nk0=Pkn0.Then (Uk,Nk) satisfies the estimates (93) and (94), and for any t≥0, we find that (uk,nk)=(Uk,Nk)+(Pkξ,Pkζ) is the unique global solution of (82).
And then, we will investigate (99)-(100) for fixed ω. Firstly, by (94), for any T>0, we can know that (102)P⋃L=1∞⋂l=1∞∩k=l∞sup0≤t≤T⁡Uk,NkV12≥L=0.We let (103)Ω~=⋃L=1∞⋂l=1∞∩k=l∞sup0≤t≤T⁡Uk,NkV12≤L.Then P(Ω∖Ω~)=0. Thus, for any fixed ω∈Ω~, there exists r(ω) with 0<r(ω)<∞ such that (104)sup0≤t≤T⁡Uk,NkV12≤rω.Then we can extract a subsequence of (Uk,Nk), which are still denoted by (Uk,Nk), such that (105)Uk⇀∗UinL∞0,T;H2αD∩H0αD,Nk⇀∗NinL∞0,T;H01D.These convergences are sufficient to pass the limit k→∞ in linear terms; however, in fact, for nonlinear terms, we need a strong convergence of Uk. From (100) and the estimate (104), it is easy to get Utk∈L∞(0,T;L2(D)). Further, we can extract a subsequence of Uk which is still denoted by Uk such that (106)Uk⟶Ustrongly  inL∞0,T;H01D.
For the nonlinear term, we can pass the limitk→∞ by a standard procedure. Thus, we prove that (U,N)∈L∞(0,T;V1) is a weak solution of (107)iUt--ΔαU-nu+iδU=f,Nt+u2x+βN=gwith initial conditions (108)U0=u0,N0=n0,x∈D.
Then(u,n)=(U,N)+(ξ,ζ) is a solution of (1)-(2) and satisfies the estimates given in Section 3.2. The continuity of the solution can be proved in the following. For any ω∈Ω~,(u2)x∈L∞(0,T;H10(D)), it is obvious that Nt=-(u2)x-βN+g∈H01(D). Then, we obtain that there is N∈C(0,T;H01(D)) for almost all ω∈Ω by Lemma 3 in [23]. Noticing Ut∈L∞(0,T;L2(D)) almost surely and applying similar methods, we can get U∈C(0,T;H2α(D)∩H0α(D)) according to [21]. So we get (u,n)∈C(0,T;V1) almost surely by definition of U and N. Thus, the solution (u,n) is continuous from [0,T] to V1 almost surely.
Because the noise is additive, we can use the same approach as [7]. Therefore, the solution (u,n) is unique almost surely in L∞(0,T;V1).Theorem 8.
If(u0,n0)∈V2,q2∈H2(D),q1∈H3α(D), and f,g∈H2(D), then there exists a unique solution (u,n)∈(R+;V2) almost surely satisfy (1)-(2). Moreover, (u,n) is continuous from R+ to V2.As a matter of fact, from Theorems7 and 8, a continuous random dynamical system can be defined in V1 and V2, respectively. Then we can, respectively, construct a random attractor endowed with the weak topology for the continuous random dynamical system in V1 and V2.The proof of Theorem8 is similar to that of Theorem 7. Here we omit the detail of the proof.


## 5. Proofs of Theorems10 and 11

In this section, the asymptotic behavior of solution for the corresponding problem is studied. We will construct a random attractor for stochastic fractional long-short wave equations inv1 equipped with the weak topology. In [13–16], we can obtain some basic concepts related to random attractors for random dynamical systems.Motivated by [13, 14], we can find the following existence result for a random attractor for a continuous RDS. It is a sufficient condition for the existence of random attractors.Theorem 9 (see [13, 14]).
AssumeΦ is a RDS on a Polish space (E,d) and there exists a random compact set K(ω) absorbing every bounded deterministic set D⊂E. Then we define a global random attractor for RDS Φ by (109)Aω=⋂τ≥0⋃t≥τΦt,θ-tω,Kθ-tω¯.Next step, according to Theorem9 and the priori estimates in Section 3, we research the random attractors for the stochastic long-short wave equations in V1 and V2.Theorem 10.
If(u0,n0)∈V1,q1∈H1(D),q2∈H2α(D), and f,g∈H1(D), then (1)-(2) have a global random weak attractor A(ω) which is a random tempered compact set in V1 endowed with the weak topology.Proof.
On the basis of the former analysis, we can consider the properties of solution a(U,N) of the system and it (107) has a unique solution (U,N)∈C(0,T;V1) for almost all ω∈Ω. Noticing that system (107) has coefficients driven by θt, a random dynamical system can be defined by (U,N) on V1. Thus, (u,n)=(U+ξ,N+ζ) also defines a continuous random dynamical system on V1, which is denoted by Φ(t,ω), and Φ(t,ω) is weakly continuous almost surely on V1. B(0,r) denotes the ball center at 0 with radius r in V1. Using estimates made in accordance with Section 3, there is a random variable R(ω) such that, for any r>0, (u,n)∈B(0,r). Therefore, there exists a random time tr(ω)>0, such that, for all t>tr(ω) and almost all ω∈Ω, (110)Φt,θ-tωu0,n0V1≤Rω.We define the random attractor A(ω) as follows: (111)Aω=⋂τ≥0⋃t≥τΦt,θ-tω,Kθ-tω¯V1ω,where the closure is taken with respect to the weak topology of V1. Next we prove that A(ω) is tempered. According to the estimates obtained in Section 3, we obtain (112)Esup0≤t≤1⁡R2θtω<∞.Then, from Birkhoff’s ergodic Theorem [24], (113)lims→±∞⁡supt∈0,1⁡R2θt+sωs=0on a θ-invariant subset of Ω with full probability measure; that is, R(ω) is tempered. So we get that A(ω) is tempered. The proof is completed.Theorem 11.
If(u0,n0)∈V2,q1∈H2(D),q2∈H3α(D), and f,g∈H2(D), then (1)-(2) have a global random weak attractor A(ω) which is a random tempered compact set in V2 endowed with the weak topology.Based on Theorem10, we can prove Theorem 11 using the same methods and ideas of proof. More precisely, there exists a random attractor for the stochastic fractional long-short wave equations in V2.Theorem 12.
If(u0,n0)∈V2,q1∈H2(D),q2∈H4α(D), then (1)-(2) have one stationary measure on V1 and V2.Proof.
If(u0,n0)∈V2, (1)-(2) has a unique solution (u,n) with (u(0),n(0))=(u0,n0) by the results given in Sections 3 and 4, which, for any t>0, satisfies (114)E-Δαn2+-Δαux2≤C,for a positive constant C>0 which is independent of t>0.
Letμt be the distribution of (nt,n,E)fort≥0. According to the classical Bogolyubov-Krylov argument [22], we define (115)μ-t=1t∫0tμsdsas (116)μ-tΓ=1t∫0tμsΓdsfor any Borel set Γ of V1: namely, Γ∈B(V1). From (114), we obtain (117)∫V1u,nV22μ-tdv=1t∫0tEus,nsV22ds≤C.According to Chebyshev’s inequality and the fact that V2 has a compact embedding into V1,{μ-t}t≥0 is tight in V1. Then there exists a sequence {μ-tk} with tk→∞ as k→∞ and a probability measure μ on V1 such that {μ-tk}→μ weakly as k→∞. Thus, using the standard argument as in [24], μ is a stationary measure for stochastic fractional long-short wave equations on V1. In addition, from (114), μ is in fact supported on V2; that is to say, μ is a stationary measure for stochastic fractional long-short wave equations on V2. The proof is completed.



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