# Global Existence for Stochastic Strongly Dissipative Zakharov Equations

Authors: Xueqin Wang; Yadong Shang; Chunlin Lei
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

The stochastic strongly dissipative Zakharov equations with white noise are studied. On the basis of the time uniform a priori estimates, we prove the existence and uniqueness of solutions in energy spacesE1 and E2, by using the standard Galerkin approximation method of stochastic partial differential equations.


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## Body

## 1. Introduction

The Zakharov system, derived by Zakharov in 1972 [1], describes the interaction between Langmuir (dispersive) and ion acoustic (approximately nondispersive) waves in an unmagnetized plasma. The usual Zakharov system defined in the space ℝd+1 is given by
(1)iEt+ΔE=nE,ntt−Δn=ΔE2,where the wave fields Ex,t and nx,t are complex and real, respectively. It has become commonly accepted that the Zakharov system is a general model to govern interaction of dispersive nondispersive waves.In the past decades, the Zakharov system has been studied by many authors [2–10]. In [4], the authors established globally in time existence and uniqueness of smooth solution for a generalized Zakharov equation in a two-dimensional case for small initial date and proved global existence of smooth solution in one spatial dimension without any small assumption for initial data. In [7, 8], the uniqueness and existence of the global classical solution of the periodic initial value problem for the system of Zakharov equations and the system of generalized Zakharov equations have been proved.In order to better qualitative agreement, it is necessary to include damping effects or effects of the loss of energy. In a realistic physical system, dissipation must be included into each equation. In [11], the author studied the following dissipative Zakharov equations:
(2)1λ2ntt−nxx−Exx2−αnt=f,(3)iEt+Exx−nE+iγE=g,where positive constants α and γ are damping coefficients and f and g are external forces. The author obtained the long time behavior of (2) and (3) on a bounded interval with initial conditions and homogeneous Dirichlet boundary conditions. The asymptotic behaviors of the solution for (2) and (3) in 1D-2D have been investigated (see [11–13]).In [14], the authors considered the following strongly dissipative Zakharov equations:
(4)1λ2ntt−nxx−Exx2−αntxx=f,(5)iEt+Exx−nE+iγE=g.They studied the Cauchy problem of (4) and (5) and proved the existence of the maximal attractor.In recent years, the importance of taking random effects into account in modeling, analyzing, simulating, and predicting complex phenomena has been widely recognized in geophysical and climate dynamics, materials science, chemistry, biology, and other areas [15–18]. Stochastic partial differential equations are appropriate mathematical models for complex systems under random influences or noise. Usually, the noise can be regarded as a simple approximation of turbulence in fluids.In [19], stochastic dissipative Zakharov equations have been studied on a regular domain D in the Itô sense. The authors proved the existence and uniqueness of solutions. Then, a global random attractor was constructed. Further, the existence of a stationary measure was proved. In [20], the existence and uniqueness of solutions are obtained. Moreover, the asymptotic behaviors of the solutions for the stochastic dissipative quantum Zakharov equations with white noise were also investigated.In the present paper, we consider the following random forced strongly dissipative Zakharov equations on a regular domainD in the Itô sense:
(6)1λ2ntt−nxx−Exx2−αntxx=f+Ẇ1,iEt+Exx−nE+iγE=g+Ẇ2,where the parameters γ>0 and α>0, and W1 and W2 are independent L2D-valued Wiener processes, which will be detailed in the next section. Here, Ẇ denotes the general derivative of the Wiener processes W with respect to time.The rest of this paper is organized as follows. In Section2, some functional setting and some conditions are given. In Section 3, a series of time uniform a priori estimates are given in different energy spaces. Some of the technique of estimates and Itô’s formula are used frequently. In Section 4, we obtain the existence and uniqueness of solutions for the stochastic strongly dissipative Zakharov equations by using the standard Galerkin approximation method in different investigated spaces. Various positive constants are denoted by C throughout this paper. Now, we state the main results of the paper.


## 2. Preliminary

In this section, we give a detailed description of the stochastic strongly dissipative Zakharov equations. LetD=0,l with 0<l<∞. Consider the following strongly dissipative stochastic Zakharov equations on a regular domain D:
(7)1λ2ntt−nxx−Exx2−αntxx=f+Ẇ1,(8)iEt+Exx−nE+iγE=g+Ẇ2,t∈R+,x∈D,with initial conditions
(9)E∣t=0=E0x,n∣t=0=n0x,nt∣t=0=n1x,x∈D,and Dirichlet boundary conditions
(10)E=0,n=0,t∈R+,x∈∂D;here, f, g, α, γ, and λ are given, with α>0 and γ>0.As in [14], to study the solution of strongly dissipative stochastic Zakharov equations, we set m=nt+εn, where ε>0 is small enough and will be chosen later. So we have the following equations:
(11)m=nt+εn,(12)1λ2mt−ελ2m−αmxx+ε2λ2n−1−εαnxx−Exx2=f+Ẇ1,(13)iEt+Exx−nE+iγE=g+Ẇ2,t∈R+,x∈D,(14)E∣t=0=E0x,n∣t=0=n0x,m∣t=0=n1x+εn0x=m0x,x∈D,(15)E=0,n=0,m=0,t∈R+,x∈∂D.Here, we give a complete probability spaceΩ,F,Ftt≥0,ℙ. The expectation operator with respect to ℙ is denoted by E. The stochastic terms W1t and W2t on Ω,F,Ftt≥0,ℙ are defined by
(16)W1t=q1xω1t,W2t=q2xω2t,where ω1t is a standard real-valued Wiener process and ω2t is a standard complex-valued Wiener process independent of ω1t. In addition, q1x and q2x are sufficiently smooth functions.We will work on the usual functional spacesL2D, HmD, and H01D. The inner product on L2D will be denoted by u,v=Re∫Duxv¯xdx,u,v∈L2D and the norm by u0=uL2=∫Du2dx1/2. The general p-norm of LpDp≥1 is denoted by uLp=∫Dupdx1/p. And the norm on HmD is denoted by um=∑0≤α≤mDαuL2D21/2. The real and imaginary parts of a complex number A are denoted, respectively, by ReA and ImA.Now, we define spacesEi,i=0,1,2, as E0=H−1D×L2D×H01D, E1=L2D×H01D×H2D∩H01D, and E2=H01D×H2D∩H01D×u∈H3D∩H01D,△u∈H01D. Endow Ei with the usual product norm ·Ei. Then, E2⊂E1⊂E0 with compact embeddings.In our approach, we need the following lemmas. LetX,·X⊂Y,·Y⊂Z,·Z be three reflective Banach spaces and X⊂Y with compact and dense embedding. Define the Banach space
(17)G=v:v∈L20,T;X,dvdt∈L20,T;Z,with
(18)vG2=∫0TvsX2ds+∫0TdvdtsZ2ds,v∈G.Then, we have the following lemma about compactness result (see [21]).Lemma 1.
IfK is bounded in G, then K is precompact in L20,T;Y.
Another lemma is needed for some maximal estimates on the stochastic integral. AssumeU and H are separable Hilbert spaces, W is a Q-Wiener process on U0 with U0=Q1/2U. And let L20=L20U0,H be the space of Hilbert-Schmidt operators from U0 to H. We have the following results (see [22]).Lemma 2.
For anyr≥1 and any L20-valued predictable process Φt,t∈0,T, we have
(19)E∫0tΦsdWs=0,Esups∈0,t∫0sΦσdWσ2r≤crsups∈0,tE∫0sΦσdWσ2r≤CrE∫0tΦsL202dsr,where cr and Cr are some positive constants dependent on r.


## 3. Time Uniform A Priori Estimates

In this section, in the sense of expectation, we give a priori estimates fornt,n,E in different spaces E0, E1, and E2.Lemma 3.
Assumeg,E0,q2∈L2D. Then, for any T>0 and p≥1, E∈L2pΩ;L∞0,T;L2D∩L∞0,∞;L2pΩ,L2D.Proof.
Applying Itô’s formula toE02, by (13), we have
(20)ddtE02=2Re∫DEtE¯dx+q202=−2γE02+2Im∫DgE¯dx+2Im∫DẆ2E¯dx+q202.Integrating (20) from 0 to t, by Young inequality, we obtain
(21)E02=E002−2γ∫0tEs02ds+2∫0tIm∫DgE¯dxds+2∫0tIm∫DẆ2E¯dxds+q202t≤E002−2γ∫0tEs02ds+2∫0tg0E0ds+2∫0tIm∫DẆ2E¯dxds+q202t≤E002−γ∫0tEs02ds+1γg02+q202t+2∫0tIm∫DẆ2E¯dxds.Taking expectation on both sides of (21), by Lemma 2, we get
(22)EE02≤EE002−γE∫0tEs02ds+1γg02+q202t.Then, by Gronwall inequality, we get(23)EE02≤e−γtEE002+1γ2g02+1γq202≤C,where C is independent of T.By (23), we obtain E∈L∞0,∞;L2Ω,L2D.Taking the supremum and expectation on both sides of (21), by Lemma 2 and (23), for any T>0, there exists a positive constant CT; we have
(24)Esup0≤t≤TE02≤EE002+Esup0≤t≤T∫0tIm∫DẆ2E¯dxds2+1γg02+q202T+1≤EE002+Cq202E∫0TE02ds+1γg02+q202T+1≤CTEE004+g04+q204+1≤CTE0,g,q2,where CTE0,g,q2 is a constant depending on E00,g0,q20 and T.By (24), we obtain E∈L2Ω;L∞0,T;L2D.By the above estimates, we can further give an estimate ofE02p for any p≥1. Now applying Itô’s formula, Young inequality, and Hölder inequality, we have
(25)ddtE02p=pE02p−1ddtE02+2pp−1E02p−1q202=pE02p−12Re∫DEtE¯dx+q202+2pp−1E02p−1q202=pE02p−1−2γEt02+2pE02p−1Im∫DgE¯dx+2pE02p−1Im∫DẆ2E¯dx+p2p−1E02p−1q202≤−2γpE02p+2pE02p−1g0E0+2pE02p−1Im∫DẆ2E¯dx+p2p−1E02p−1q202≤−γpE02p+pγE02p−1g02+2pE02p−1Im∫DẆ2E¯dx+p2p−1E02p−1q202≤−γpE02p+γp2E02p+2pE02p−1Im∫DẆ2E¯dx+Cg0,q20=−γp2E02p+2pE02p−1Im∫DẆ2E¯dx+Cg0,q20.Integrating (25) from 0 to t, we obtain
(26)E02p≤E002p−γp2∫0tE02pds+Cg0,q20t+2p∫0tE02p−1Im∫DẆ2E¯dxds.Taking expectation on both sides of (26), we get
(27)EE02p≤EE002p−γp2E∫0tE02pds+Cg0,q20t.Then, by Gronwall inequality, we get(28)EE02p≤e−γpt/2EE002p+Cg0,q20≤Cg,q2,where Cg,q2 is independent of T.By (28), we obtain E∈L∞0,∞;L2pΩ,L2D.Taking the supremum and expectation on both sides of (26), by Lemma 2 and (28), for any T>0, there exists a positive constant CT; we have
(29)Esup0≤t≤TE02p≤EE002p+Cg0,q20T+1+p2Esup0≤t≤T∫0tE02p−1Im∫DẆ2E¯dxds2≤EE002p+Cg0,q20T+1+Cp2q202E∫0TE04p−2ds≤CTE00,g0,q20≤CTE0,g,q2,where CTE0,g,q2 is a constant depending on E00,g0,q20 and T.By (29), we obtain E∈L2pΩ;L∞0,T;L2D.Lemma3 is proved completely.Lemma 4.
Assumen1,n0,E0∈E0,f,q1∈H−1D and g,q2∈H01D. Then, for any T>0 and p≥1, nt,n,E∈L2pΩ;L∞0,T;E0∩L∞0,∞;L2pΩ,E0.Proof.
Applying Itô’s formula tom−12, by (12), we have
(30)12λ2ddtm−12=∫D1λ2mt−Δ−1mdx+λ22q1−12=ελ2m−12−αm02−ε2λ2∫D−Δ−1/2n−Δ−1/2mdx−1−εα∫Dnmdx−∫DE2mdx+∫Df−Δ−1mdx+∫DẆ1−Δ−1mdx+λ22q1−12.Sincem=nt+εn, from (30), we obtain
(31)ddt12λ2m−12+121−εαn02+ε22λ2n−12=ελ2m−12−αm02−ε3λ2n−12−ε1−εαn02−∫DE2mdx+∫Df−Δ−1mdx+∫DẆ1−Δ−1mdx+λ22q1−12≤ελ2−αλ1m−12−ε3λ2n−12−ε1−εαn02−∫DE2mdx+∫Df−Δ−1mdx+∫DẆ1−Δ−1mdx+λ22q1−12.where λ1 is the first eigenvalue of −Δ.By (31), we get
(32)ddt12λ2m−12+121−εαn02+ε22λ2n−12+αλ1−ελ2m−12+ε1−εαn02+ε3λ2n−12≤−∫DE2mdx+∫Df−Δ−1mdx+λ22q1−12+∫DẆ1−Δ−1mdx.Applying Itô’s formula toEx02, by (13), we have
(33)ddtEx02=2Re∫DExE¯txdx+q212.SubstitutingExx=−iEt+nE−iγE+g+Ẇ2 into (33), we have
(34)ddtEx02=−2Re∫DExxE¯tdx+q212=−2Re∫DnE−iγE+g+Ẇ2E¯tdx+q212=−2Re∫DnEE¯tdx+2Re∫DγE¯−iEtdx−2Re∫DgE¯tdx−2Re∫DẆ2E¯tdx+q212=−2Re∫DnEE¯tdx−2γEx02−2γ∫DnE2dx−2Re∫DγE¯gdx−2Re∫DγE¯Ẇ2dx−2Re∫DgE¯tdx−2Re∫DẆ2E¯tdx+q212=−2Re∫DnEE¯tdx−2γEx02−2γ∫DnE2dx+q212−2Re∫DγE¯+E¯tgdx−2Re∫DγE¯+E¯tẆ2dx.In addition, applying Itô’s formula to∫DnE2dx, we have
(35)ddt∫DnE2dx=∫DntE2dx+2Re∫DnEE¯tdx+∫Dnq22dx.Substituting (35) into (34), we obtain
(36)ddtEx02+∫DnE2dx=−2γEx02−2γ∫DnE2dx+q212−2Re∫DγE¯+E¯tgdx−2Re∫DγE¯+E¯tẆ2dx+∫DntE2dx+∫Dnq22dx=−2γEx02−2γ+ε∫DnE2dx+q212+∫Dnq22dx−2Re∫DγE¯+E¯tgdx−2Re∫DγE¯+E¯tẆ2dx+∫DmE2dx.Taking0<ε≤min1/α,αλ1λ2/2, and by (32) and (36), we have
(37)ddt12λ2m−12+121−εαn02+ε22λ2n−12+ddtEx02+∫DnE2dx+ελ2m−12+ε1−εαn02+ε3λ2n−12+2γEx02+2γ+ε∫DnE2dx≤12ddt1λ2m−12+1−εαn02+ε2λ2n−12+ddtEx02+∫DnE2dx+αλ1−ελ2m−12+ε1−εαn02+ε3λ2n−12+2γEx02+2γ+ε∫DnE2dx≤∫Df−Δ−1mdx+∫DẆ1−Δ−1mdx+λ22q1−12+q212−2Re∫DγE¯+E¯tgdx−2Re∫DγE¯+E¯tẆ2dx+∫Dnq22dx.By Hölder inequality and Young inequality, we have the following estimates:(38)−2Re∫DγE¯+E¯tgdx≤2Ex0gx0+2gL∞n0E0−2Re∫DiẆ2g¯dx≤γ2Ex02+2γgx02+C1gx04+ε1−εα6n02+C2E04+2Im∫DẆ2g¯dx,γ+ε∫DnE2dx≤Cn0EL42≤Cn0E03/2Ex01/2≤ε1−εα6n02+γ2Ex02+C3E06,∫Df−Δ−1mdx≤f−1m−1≤ε2λ2m−12+λ22εf−12,∫Dnq22dx≤n0q2L42≤n0Cq203/2q2x01/2≤ε1−εα6n02+Cq2x02+q206.By (38), we obtain
(39)ddt12λ2m−12+121−εαn02+ε22λ2n−12+ddtEx02+∫DnE2dx+ελ2m−12+ε1−εαn02+ε3λ2n−12+2γEx02+2γ+ε∫DnE2dx≤∫Df−Δ−1mdx+∫DẆ1−Δ−1mdx+λ22q1−12+q212−2Re∫DγE¯+E¯tgdx−2Re∫DγE¯+E¯tẆ2dx+∫Dnq22dx≤ε2λ2m−12+λ22εf−12+∫DẆ1−Δ−1mdx+λ22q1−12+q212+γ2Ex02+2γgx02+C1gx04+ε1−εα6n02+C2E04+2Im∫DẆ2g¯dx−2Re∫DγE¯+E¯tẆ2dx+ε1−εα6n02+Cq212+q206.By (39), we get
(40)ddt12λ2m−12+121−εαn02+ε22λ2n−12+ddtEx02+∫DnE2dx+ελ2m−12+ε1−εαn02+ε3λ2n−12+2γEx02+γ∫DnE2dx≤ε2λ2m−12+λ22εf−12+∫DẆ1−Δ−1mdx+λ22q1−12+q212+γ2Ex02+2γgx02+ε1−εα6n02+C2E04+C1gx04+2Im∫DẆ2g¯dx−2Re∫DγE¯+E¯tẆ2dx+ε1−εα6n02+Cq212+q206+ε1−εα6n02+γ2Ex02+C3E06.Now, takingβ=minε,γ and letting H0t=1/2λ2m−12+1−εα/2n02+ε2/2λ2n−12+Ex02+∫DnE2dx, the above inequality is changed into
(41)ddtH0t+βH0t≤ddt12λ2m−12+121−εαn02+ε22λ2n−12+ddtEx02+∫DnE2dx+ε2λ2m−12+ε1−εα2n02+ε3λ2n−12+γEx02+γ∫DnE2dx≤Cf−1,g1,q1−1,q21+∫DẆ1−Δ−1mdx+CE06+2Im∫DẆ2g¯dx−2Re∫DγE¯+E¯tẆ2dx≤Cf,g,q1,q2+∫DẆ1−Δ−1mdx+CE06+2Im∫DẆ2g¯dx−2Re∫DγE¯+E¯tẆ2dx.Integrating (41) from 0 to t, we obtain
(42)H0t−H00+β∫0tH0sds≤C∫0tEs06ds+Cf,g,q1,q2t+∫0t∫DẆ1−Δ−1mdxds+2∫0tIm∫DẆ2g¯dxds−2∫0tRe∫DγE¯+E¯tẆ2dxds.Taking expectation on both sides of the above inequality, by (28), we get
(43)EH0t−EH00+βE∫0tH0sds≤CE∫0tEs06ds+Cf,g,q1,q2t≤CE0,f,g,q1,q2t.By (23) and Gronwall inequality, we have
(44)EH0t≤e−βtEH00+CE0,f,g,q1,q2≤CE0,f,g,q1,q2,where CE0,f,g,q1,q2 is independent of T.Since(45)∫DnE2dx≤Cn0EL42≤Cn0E03/2Ex01/2≤1−εα4n02+12Ex02+CE06,we have
(46)Cm−12+n02+n−12+Ex02−CE06≤H0t≤Cm−12+n02+n−12+Ex02+CE06.By (41) and (46), we have
(47)ddtH0t≤Cf,g,q1,q2+∫DẆ1−Δ−1mdx+CE06+2Im∫DẆ2g¯dx−2Re∫DγE¯+E¯tẆ2dx.Therefore, by (28) and (44), we have
(48)Em−12+n02+n−12+Ex02≤CEH0t+CEE06≤Ce−βtEH00+CE0,f,g,q1,q2+Ce−3γt/2EE006+Cg0,q20≤CE0,f,g,q1,q2,where CE0,f,g,q1,q2 is independent of T.By (48), we obtain nt,n,E∈L∞0,∞;L2Ω,E0.Further, we give an estimate ofH0pt for any p≥1. First as
(49)nE−12=∫D−Δ−1nEnE¯dx≤−Δ−1nEL∞n0E0≤C−Δ−1nE1n0E0≤CnE−1n0E0,by (47), we have
(50)nE−12≤Cn02E02≤CH0t+E06E02,(51)m−12+n02+n−12+Ex02≤CH0t+E06.Applying Itô’s formula toH0pt, by (41), (50), (51), and Young inequality, we obtain
(52)ddtH0pt=pH0p−1tddtH0t+12pp−1CH0p−2tE12q212+g02q202+12pp−1CH0p−2tnE−12q212+m−12q2−12≤pH0p−1t−βH0t+Cf,g,q1,q2+CE06+pH0p−1t∫DẆ1−Δ−1mdx+2Im∫DẆ2g¯dx+pH0p−1t−2Re∫DγE¯+E¯tẆ2dx+12pp−1CH0p−2tE12q212+g02q202+12pp−1CH0p−2tnE−12q212+12pp−1CH0p−2tm−12q2−12≤−βpH0pt+pH0p−1tCf,g,q1,q2+CE06+pH0p−1t∫DẆ1−Δ−1mdx+2Im∫DẆ2g¯dx+pH0p−1t−2Re∫DγE¯+E¯tẆ2dx+12pp−1CH0p−2t2H0t+CE06q212+12pp−1CH0p−2tg02q202+12pp−1CH0p−2tH0t+E06CE02q212+12pp−1CH0p−2t2λ2H0t+CE06q2−12≤−βp2H0pt+CE0,f,g,q1,q2+CE06p+E03p+pH0p−1t∫DẆ1−Δ−1mdx+2Im∫DẆ2g¯dx+pH0p−1t−2Re∫DγE¯+E¯tẆ2dx.Integrating from 0 tot and taking expectation on both sides of the above inequality, by Hölder inequality and (28), we derive
(53)EH0pt≤EH0p0−βp2E∫0tH0psds+CE0,f,g,q1,q2t.By Gronwall inequality, we have(54)EH0pt≤e−βpt/2EH0p0+CE0,f,g,q1,q2≤CE0,f,g,q1,q2,n0,n1,where CE0,f,g,q1,q2,n0,n1 is independent of T.By (28), (51), and (54), we obtain nt,n,E∈L∞0,∞;L2pΩ,E0.On the other hand, integrating from 0 tot on both sides of (47), deduce
(55)H0t≤H00+Cf,g,q1,q2t+∫0t∫DẆ1−Δ−1mdxds2+C∫0tE06ds+∫0tIm∫DẆ2gdxds2+∫0tRe∫DγE¯+E¯tẆ2dxds2+C.Now, taking the supremum and expectation on both sides of the above inequality, we have(56)Esup0≤t≤TH0t≤EH00+Cf,g,q1,q2T+C+Esup0≤t≤T∫0t∫DẆ1−Δ−1mdxds2+CEsup0≤t≤T∫0tE06ds+Esup0≤t≤T∫0tIm∫DẆ2gdxds2+Esup0≤t≤T∫0tRe∫DγE¯+E¯tẆ2dxds2≤EH00+Cf,g,q1,q2T+Cq1−12E∫0Tm−12dt+CE∫0TE06ds+Cq202E∫0Tg02dt+C+Esup0≤t≤T∫0tRe∫D−iE¯xx+inE¯+ig¯+iẆ2¯Ẇ2dxds2≤EH00+Cf,g,q1,q2T+Cq1−12E∫0Tm−12dt+CE∫0TE06ds+Cq202E∫0Tg02dt+C+Cq212E∫0TEx02dt+Cq212E∫0TnE−12dt≤EH00+Cf,g,q1,q2T+C+CE∫0TE06ds+Cq202E∫0Tg02dt+Cq1−12E∫0T2λ2H0t+CE06dt+Cq212E∫0T2H0t+CE06dt+Cq212E∫0TCH0t+E06E02dt.Then, by (28), (54), and (56), we obtain
(57)Esup0≤t≤TH0t≤CTE0,f,g,q1,q2,n1,n0,where CT depends on the initial data.Therefore, by (48), we have
(58)Esup0≤t≤Tnt,n,EE02≤CTE0,f,g,q1,q2,n1,n0.By (58), we obtain E∈L2Ω;L∞0,T;E0.In addition, from (52), we have
(59)ddtH0pt≤CE0,f,g,q1,q2+CE06p+E03p+pH0p−1t∫DẆ1−Δ−1mdx+2Im∫DẆ2g¯dx+pH0p−1t−2Re∫DγE¯+E¯tẆ2dx.Integrating from 0 tot and taking the supremum and the expectation on both sides of (59), we have
(60)Esup0≤t≤TH0pt≤CE0,f,g,q1,q2T+CEsup0≤t≤T∫0tE06p+E03pds+pH0p−1t∫DẆ1−Δ−1mdx+2pH0p−1tIm∫DẆ2g¯dx+pH0p−1t−2Re∫D−iE¯xx+inE¯+ig¯+iẆ2¯Ẇ2dx≤CE0,f,g,q1,q2T+Esup0≤t≤T∫0tE06p+E03pds+Esup0≤t≤T∫0tH0p−1t∫DẆ1−Δ−1mdxds2+Esup0≤t≤T∫0tH0p−1tIm∫DẆ2g¯dxds2+C+Esup0≤t≤T∫0tH0p−1tIm∫DE¯xxẆ2dxds2+Esup0≤t≤T∫0tH0p−1tIm∫DnE¯Ẇ2dxds2.By (50), (51), Hölder inequality, and Young inequality, we have the following estimates:
(61)CEsup0≤t≤T∫0tE06p+E03pds≤CE∫0TE06p+E03pds,Esup0≤t≤T∫0tH0p−1t∫DẆ1−Δ−1mdxds2≤Cq1−12E∫0TH02p−2tm−12ds≤Cq1−12E∫0TH02p−2tCH0t+E06ds≤Cq1−12E∫0TH02pt+E06p+1ds,Esup0≤t≤T∫0tH0p−1tIm∫DẆ2g¯dxds2≤Cq202E∫0TH02p−2tg02ds≤Cq202E∫0TH02pt+g02pds≤Cq202E∫0TH02pt+1ds,Esup0≤t≤T∫0tH0p−1tIm∫DE¯xxẆ2dxds2≤Cq2x02E∫0TH02p−2tEx02ds≤Cq2x02E∫0TH02p−2tCH0t+E06ds≤Cq2x02E∫0TH02pt+E06p+1ds,Esup0≤t≤T∫0tH0p−1tIm∫DnE¯Ẇ2dxds2≤Cq212E∫0TH02p−2tnE−12ds≤Cq212E∫0TH02p−2tCH0t+E06E02ds≤Cq212E∫0TH02pt+E04p+E08pds.By (28), (54), (60), and (61), we have
(62)Esup0≤t≤TH0pt≤CEH0p0+CTE0,f,g,q1,q2,n1,n0≤CTE0,f,g,q1,q2,n1,n0.Moreover, by (29), (51), and (62), we have
(63)CEsup0≤t≤Tm−12p+n02p+n−12p+Ex02p≤CEsup0≤t≤TCH0t+E062p≤CEsup0≤t≤TH02pt+E012p≤CTE0,f,g,q1,q2,n1,n0,where CT depends on the initial data.Therefore, by (63), we have
(64)Esup0≤t≤Tnt,n,EE02p≤CTE0,f,g,q1,q2,n1,n0.By (64), we obtain E∈L2pΩ;L∞0,T;E0.Lemma4 is proved completely.Lemma 5.
Assumen1,n0,E0∈E1, f∈L2D, q1∈H01D, and g,q2∈H2D∩H01D. Then, for any T>0 and p≥1, nt,n,E∈L2pΩ;L∞0,T;E1∩L∞0,∞;L2pΩ,E1.Proof.
Applying Itô’s formula tom02, by (12), we have
(65)12λ2ddtm02=∫D1λ2mtmdx+λ22q102=ελ2m02−αm12−ε2λ2∫Dnmdx+1−εα∫Dnxxmdx+∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102.Sincem=nt+εn, from (65), we obtain
(66)ddt12λ2m02+ε22λ2n02+1−εα2nx02=ελ2m02−αm12−ε3λ2n02−ε1−εαnx02+∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102≤ελ2−αλ1m02−ε3λ2n02−ε1−εαnx02+∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102,where λ1 is the first eigenvalue of −Δ. By (66), we obtain
(67)ddt12λ2m02+ε22λ2n02+1−εα2nx02−ελ2−αλ1m02+ε3λ2n02+ε1−εαnx02≤∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102.Applying Itô’s formula toExx02, by (13), we have
(68)ddtExx02=2Re∫DExxE¯txxdx+q222.SubstitutingExx=−iEt+nE−iγE+g+Ẇ2 into (68), we have
(69)ddtExx02=2Re∫D−iEt+nE−iγE+g+Ẇ2E¯txxdx+q222=2Re∫DnEE¯txxdx+2Re∫DgE¯txxdx+2Re∫DẆ2E¯txxdx+q222+2Re∫DγE¯xx−Exx+nE−iγE+g+Ẇ2dx=2Re∫DnEE¯txx+γE¯xxdx+2Re∫DgE¯txx+γE¯xxdx+2Re∫DẆ2E¯txx+γE¯xxdx−2γExx02+q222.From (66) and (69), we obtain
(70)ddt12λ2m02+ε22λ2n02+1−εα2nx02+Exx02−ελ2−αλ1m02+ε3λ2n02+ε1−εαnx02≤∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102+2Re∫DnEE¯txx+γE¯xxdx+2Re∫DgE¯txx+γE¯xxdx+2Re∫DẆ2E¯txx+γE¯xxdx−2γExx02+q222,since
(71)ddt−2Re∫DnEE¯xxdx=−Re∫DntExx2dx+2Re∫DntEx2dx−2Re∫DnEE¯txxdx−2Re∫DnEtE¯xxdx−2Re∫Dnq2q¯2xxdx.Substituting (71) into (70), we have
(72)ddt12λ2m02+ε22λ2n02+1−εα2nx02+Exx02−ελ2−αλ1m02+ε3λ2n02+ε1−εαnx02≤∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102+2Re∫DnEE¯txxdx+2γRe∫DnEE¯xxdx+2Re∫DgE¯txx+γE¯xxdx+2Re∫DẆ2E¯txx+γE¯xxdx−2γExx02+q222≤∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102−Re∫DntExx2dx+2Re∫DntEx2dx+ddt2Re∫DnEE¯xxdx−2Re∫DnEtE¯xxdx−2Re∫Dnq2q¯2xxdx+2γRe∫DnEE¯xxdx+2Re∫DgE¯txx+γE¯xxdx+2Re∫DẆ2E¯txx+γE¯xxdx−2γExx02+q222.Taking0<ε≤min1/α,αλ1λ2/2, and by (72), we have
(73)ddt12λ2m02+ε22λ2n02+1−εα2nx02+ddtExx02−2Re∫DnEE¯xxdx+ελ2m02+ε3λ2n02+ε1−εαnx02+2γExx02−2γRe∫DnEE¯xxdx≤∫DExx2mdx+∫Dfmdx+∫DẆ1mdx+λ22q102−Re∫DntExx2dx+2Re∫DntEx2dx+q222−2Re∫DnEtE¯xxdx−2Re∫Dnq2q¯2xxdx+2Re∫DgE¯txx+γE¯xxdx+2Re∫DẆ2E¯txx+γE¯xxdx≤∫Dfmdx+∫DẆ1mdx+λ22q102−εRe∫DnxEx2dx+2Re∫DmEx2dx−2εRe∫DnEx2dx+q222−2Re∫DnEtE¯xxdx−2Re∫Dnq2q¯2xxdx+2Re∫DgE¯txx+γE¯xxdx+2Re∫DẆ2E¯txx+γE¯xxdx.By Hölder inequality and Young inequality, we have the following estimates:(74)∫Dfmdx≤f0m0≤Cf02+ε4λ2m02,−εRe∫DnxEx2dx+2Re∫DmEx2dx−2εRe∫DnEx2dx≤εnx0ExL42+2m0ExL42+2εn0ExL42≤εnx0+2m0+2εn0Ex03/2Exx01/2≤ε1−εα6nx02+ε4λ2m02+ε33λ2n02+γ5Exx02+CEx06,−2Re∫DnEtE¯xxdx=2Re∫DinE¯xxiEtdx=2Re∫Din2E¯xxEdx+2γRe∫DnE¯xxEdx+2Re∫DinE¯xxgdx+2Re∫DinE¯xxẆ2dx≤nL42Exx0EL∞+n0Exx0EL∞+n0Exx0gL∞−2Im∫DnE¯xxẆ2dx≤n03/2nx01/2Exx0Ex0+n0Exx0Ex0+n0Exx0gx0−2Im∫DnE¯xxẆ2dx≤Cn012+ε1−εα6nx02+γ5Exx02+CEx08+Cn04+2γ5Exx02+CEx04+Cgx04−2Im∫DnE¯xxẆ2dx≤ε1−εα6nx02+3γ5Exx02+Cn012+Ex08+Cn04+Ex04+gx04−2Im∫DnE¯xxẆ2dx,−2Re∫Dnq2q¯2xxdx≤ε33λ2n02+Cq2xx04,2Re∫DγE¯xx+E¯txxgdx=2Re∫D−iE¯xx+inE¯+ig¯+iẆ2¯xxgdx≤2Exx0gxx0+2gxL∞nx0E0+n0Ex0−2Re∫DiẆ2g¯xxdx≤γ5Exx02+Cgxx02+Cgxx04+ε1−εα6nx02+CE04+Cgxx04+ε34λ2n02+CEx04+2Im∫DẆ2g¯xxdx≤γ5Exx02+ε1−εα6nx02+ε33λ2n02+2Im∫DẆ2g¯xxdx+Cgxx02+gxx04+E04+Ex04.By (73), we have
(75)ddt12λ2m02+ε22λ2n02+1−εα2nx02+ddtExx02−2Re∫DnEE¯xxdx+ελ2m02+ε3λ2n02+ε1−εαnx02+2γExx02−2γRe∫DnEE¯xxdx≤Cf02+ε4λ2m02+∫DẆ1mdx+λ22q102+ε1−εα6nx02+ε4λ2m02+ε36λ2n02+γ5Exx02+CEx06+q222+ε1−εα6nx02+3γ5Exx02+Cn012+Ex08+Cn04+Ex04+gx04−2Im∫DnE¯xxẆ2dx+ε36λ2n02+Cq2xx04+γ5Exx02+ε1−εα6nx02+ε36λ2n02+2Im∫DẆ2g¯xxdx+Cgxx02+gxx04+E04+Ex04+2Re∫DẆ2E¯txx+γE¯xxdx.By (75), we get
(76)ddt12λ2m02+ε22λ2n02+1−εα2nx02+ddtExx02−2Re∫DnEE¯xxdx+ε2λ2m02+ε32λ2n02+ε1−εα2nx02+γExx02−2γRe∫DnEE¯xxdx≤Cf02+∫DẆ1mdx+λ22q102+CEx06+q222+Cn012+Ex08+Cn04+Ex04+gx04−2Im∫DnE¯xxẆ2dx+Cq2xx04+2Im∫DẆ2g¯xxdx+Cgxx02+gxx04+E04+Ex04+2Re∫DẆ2E¯txx+γE¯xxdx≤Cf0,q10,q22,gx0,gxx0,+CEx0,n0,E0+∫DẆ1mdx−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx.Now, takingβ=minε,γ and letting H1t=1/2λ2m02+ε2/2λ2n02+1−εα/2nx02+Exx02−2Re∫DnEE¯xxdx, by (76), the above inequality is changed into
(77)ddtH1t+βH1t≤Cf0,q10,q22,gx0,gxx0+CEx0,n0,E0+∫DẆ1mdx−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx≤Cf,g,q1,q2,E0,n0,n1+∫DẆ1mdx−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx.Integrating (77) from 0 to t and taking expectation on both sides of the above inequality, we get
(78)EH1t−EH10+βE∫0tH1sds≤Cf,g,q1,q2,E0,n0,n1t.By Gronwall inequality, we have(79)EH1t≤e−βtEH10+Cf,g,q1,q2,E0,n0,n1≤Cf,g,q1,q2,E0,n0,n1,where Cf,g,q1,q2,E0,n0,n1 is independent of T.Since(80)2Re∫DnEE¯xxdx≤2EL∞n0Exx0≤CEx0n0Exx0≤12Exx02+CEx04+n04,we have
(81)Cm02+n02+nx02+Exx02−CEx04+n04≤H1t≤Cm02+n02+nx02+Exx02+CEx04+n04.By (77) and (81), we get
(82)ddtH1t≤Cf,g,q1,q2,E0,n0,n1+∫DẆ1mdx−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx.By (28), (54), (79), and (81), we get
(83)Em02+n02+nx02+Exx02≤EH1t+CEEx04+n04≤Cf,g,q1,q2,E0,n0,n1,where Cf,g,q1,q2,E0,n0,n1 is independent of T.By (83), we obtain nt,n,E∈L∞0,∞;L2Ω,E1.Further, we give an estimate ofH1pt for any p≥1. First as
(84)nE02=∫DnEnE¯dx≤nE0n0EL∞≤CnE0n0Ex0,nExx−12=∫D−Δ−1nExxnE¯xxdx≤−Δ−1nEL∞n0Exx0≤C−Δ−1nExx1n0Exx0≤CnExx−1n0Exx0.By (81) and (84), we have
(85)nE02≤Cn02Ex02≤CH1t+Ex04+n04Ex02,(86)nExx−12≤Cn02Exx02≤CH1t+Ex04+n04Exx02,(87)m02+n02+nx02+Exx02≤CH1t+Ex04+n04.Applying Itô’s formula toH1pt, by (77), (85), (86), (87), and Young inequality, we obtain
(88)ddtH1pt=pH0p−1tddtH1t+12pp−1CH0p−2tE22q222+g12q212+nE02q222+12pp−1CH0p−2tnExx−12q212+m02q102=pH0p−1t−βH1t+Cf,g,q1,q2,E0,n0,n1+∫DẆ1mdx+pH0p−1t−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx+12pp−1CH0p−2tE22q222+g12q212+nE02q222+12pp−1CH0p−2tnExx−12q212+m02q102≤−βpH0pt+Cf,g,q1,q2,E0,n0,n1+pH0p−1t∫DẆ1mdx+pH0p−1t−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx+12pp−1CH0p−2tE22q222+g12q212+nE02q222+12pp−1CH0p−2tnExx−12q212+m02q102≤−βpH0pt+Cf,g,q1,q2,E0,n0,n1+pH0p−1t∫DẆ1mdx+pH0p−1t−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx+12pp−1CH0p−2tE22q222+12pp−1CH0p−2tg12q212+12pp−1CH0p−2tnE02q222+12pp−1CH0p−2tnExx−12q212+12pp−1CH0p−2tm02q102≤−βpH0pt+Cf,g,q1,q2,E0,n0,n1+pH0p−1t∫DẆ1mdx+pH0p−1t−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx+12pp−1CH0p−2tH1t+Ex04+n04q222+12pp−1CH0p−2tg12q212+12pp−1CH0p−2tH1t+Ex04+n04Ex02q222+12pp−1CH0p−2tH1t+Ex04+n04H1tq212+12pp−1CH0p−2tH1t+Ex04+n04Ex04q212+12pp−1CH0p−2tH1t+Ex04+n04n04q212+12pp−1CH0p−2tH1t+Ex04+n04q102≤−βp2H0pt+Cf,g,q1,q2,E0,n0,n1+pH0p−1t∫DẆ1mdx+pH0p−1t−2Im∫DnE¯xxẆ2dx+2Re∫DẆ2E¯txx+γE¯xxdx+CEx04p+Ex06p+n04p.Taking expectation on both sides of the above inequality, by Hölder inequality and (88), we derive
(89)EH1pt≤EH1p0−βp2E∫0tH1psds+CE0,f,g,q1,q2,n0,n1t.By Gronwall inequality, we have(90)EH1pt≤e−βpt/2EH1p0+CE0,f,g,q1,q2,n0,n1≤CE0,f,g,q1,q2,n0,n1,where CE0,f,g,q1,q2,n0,n1 is independent of T.By (87) and (90), we obtain nt,n,E∈L∞0,∞;L2pΩ,E1.By the same approach of Lemma4, we obtain
(91)Esup0≤t≤TH1t≤CTE0,f,g,q1,q2,n1,n0,where CT depends on the initial data. Therefore, by (47), we have
(92)Esup0≤t≤Tnt,n,EE12≤CTE0,f,g,q1,q2,n1,n0.By (92), we obtain E∈L2Ω;L∞0,T;E1.Moreover, we get(93)Esup0≤t≤TH1pt≤CEH1p0+CTE0,f,g,q1,q2,n1,n0≤CTE0,f,g,q1,q2,n1,n0.By (87), we have
(94)CEsup0≤t≤Tm02p+n02p+nx−12p+Exx02p≤CEsup0≤t≤TCH1t+Ex04+n042p≤CEsup0≤t≤TH12pt+Ex08p+n08p≤CTE0,f,g,q1,q2,n1,n0,where CT depends on the initial data. Therefore, by (94), we have
(95)Esup0≤t≤Tnt,n,EE12p≤CTE0,f,g,q1,q2,n1,n0.By (95), we obtain E∈L2pΩ;L∞0,T;E1.Lemma5 is proved completely.Lemma 6.
Assumen1,n0,E0∈E2, f∈H01D, g,q1∈H2D∩H01D, and q2∈H3D∩H01D. Then, for any T>0 and p≥1, nt,n,E∈L2pΩ;L∞0,T;E2∩L∞0,∞;L2pΩ,E2.Proof.
Applying Itô’s formula tomx02, by (12), we have
(96)12λ2ddtmx02=−∫D1λ2mtmxxdx+λ22q112=∫Dελ2m+αmxx−ε2λ2n−mxxdx+∫D1−εαnxx+Exx2+f+Ẇ1−mxxdx+λ22q112=ελ2mx02−αm22−ε2λ2∫Dnxmxdx−1−εα∫Dnxxmxxdx−∫DExx2mxxdx−∫Dfmxxdx−∫DẆ1mxxdx+λ22q112.Sincem=nt+εn, from (96), we obtain
(97)ddt12λ2mx02+ε22λ2nx02+1−εα2nxx02=ελ2mx02−αm22−ε3λ2nx02−ε1−εαnxx02−∫DExx2mxxdx+∫Dfxmxdx−∫DẆ1mxxdx+λ22q112,where λ1 is the first eigenvalue of −Δ.By (97), we obtain
(98)ddt12λ2mx02+ε22λ2nx02+1−εα2nxx02−ελ2−αλ1mx02+ε3λ2nx02+ε1−εαnxx02≤−∫DExx2mxxdx+∫Dfxmxdx−∫DẆ1mxxdx+λ22q112.Applying Itô’s formula toExxx02, by (13), we have
(99)ddtExxx02=2Re∫DExxxE¯txxxdx+q232=−2Re∫DExxE¯txxxxdx+q232.SubstitutingExx=−iEt+nE−iγE+g+Ẇ2 into (99), we have
(100)ddtExxx02=−2Re∫D−iEt+nE−iγE+g+Ẇ2E¯txxxxdx+q232=−2Re∫DnEE¯txxxxdx−2Re∫DγE¯xxxxExx+nE−iγE+g+Ẇ2dx−2Re∫DgE¯txxxxdx−2Re∫DẆ2E¯txxxxdx+q232=−2γExxx02−2Re∫DnEE¯txxxx+γE¯xxxxdx−2Re∫DẆ2E¯txxxx+γE¯xxxxdx−2Re∫DgE¯txxxx+γE¯xxxxdx+q232.From (98) and (100), we obtain
(101)ddt12λ2mx02+ε22λ2nx02+1−εα2nxx02+Exxx02−ελ2−αλ1mx02+ε3λ2nx02+ε1−εαnxx02≤−∫DExx2mxxdx+∫Dfxmxdx−∫DẆ1mxxdx+λ22q112−2γExxx02−2Re∫DnEE¯txxxx+γE¯xxxxdx−2Re∫DẆ2E¯txxxx+γE¯xxxxdx−2Re∫DgE¯txxxx+γE¯xxxxdx+q232.Taking0<ε≤min1/α,αλ1λ2/2, and by (101), we have
(102)ddt12λ2mx02+ε22λ2nx02+1−εα2nxx02+Exxx02+ελ2mx02+ε3λ2nx02+ε1−εαnxx02+2γExxx02≤−∫DExx2mxxdx+∫Dfxmxdx−∫DẆ1mxxdx+λ22q112−2Re∫DnEE¯txxxx+γE¯xxxxdx−2Re∫DẆ2E¯txxxx+γE¯xxxxdx−2Re∫DgE¯txxxx+γE¯xxxxdx+q232.Since(103)2Re∫DnEE¯txxxx+γE¯xxxxdx=2Re∫DnxxEE¯txx+γE¯xxdx+2Re∫DnExxE¯txx+γE¯xxdx+4Re∫DnxExE¯txx+γE¯xxdx,ddt∫DnxxExx2dx=∫DntxxExx2dx+2Re∫DnxxE¯tExxdx+2Re∫DnxxE¯Etxxdx−4ddtRe∫DnxExE¯xxdx+4Re∫DntxExE¯xxdx+4Re∫Dnxq2xq¯2xxdx+2Re∫Dnxxq¯2q2xxdx+2Re∫Dnxxq¯2xq2xdx,ddt∫DnExx2dx=∫DntExx2dx+2Re∫DnExxE¯txxdx+∫Dnq2xx2dx,ddt2Re∫DnxExE¯xxdx=2Re∫DntxExE¯xxdx+2Re∫DnxEtxE¯xxdx+2Re∫DnxExE¯txxdx+2Re∫Dnxq2xq¯2xxdx,then we deduce
(104)2Re∫DnEE¯txxxx+γE¯xxxxdx=ddt∫DnxxExx2dx+ddt∫DnExx2dx+ddt8Re∫DnxExE¯xxdx−∫DntxxExx2dx−2Re∫DnxxE¯tExxdx−4Re∫DntxExE¯xxdx−∫DntExx2dx+2γRe∫DnxxEE¯xxdx−4Re∫DntxExE¯xxdx−4Re∫DnxEtxE¯xxdx+2γRe∫DnExxE¯xxdx+4γRe∫DnxExE¯xxdx−4Re∫Dnxq2xq¯2xxdx−∫Dnq2xx2dx−4Re∫Dnxq2xq¯2xxdx−2Re∫Dnxxq¯2q2xxdx−2Re∫Dnxxq¯2xq2xdx.Substituting (104) into (102), we have
(105)ddt12λ2mx02+ε22λ2nx02+1−εα2nxx02+Exxx02ddt∫DnxxExx2dx+∫DnExx2dx+8Re∫DnxExE¯xxdx+ελ2mx02+ε3λ2nx02+ε1−εαnxx02+2γExxx02≤−∫DExx2mxxdx+∫Dfxmxdx−∫DẆ1mxxdx+λ22q112+∫DntxxExx2dx+2Re∫DnxxE¯tExxdx+4Re∫DntxExE¯xxdx+∫DntExx2dx−2γRe∫DnxxEE¯xxdx+4Re∫DntxExE¯xxdx+4Re∫DnxEtxE¯xxdx−2γRe∫DnExxE¯xxdx−4γRe∫DnxExE¯xxdx+4Re∫Dnxq2xq¯2xxdx+∫Dnq2xx2dx+4Re∫Dnxq2xq¯2xxdx+2Re∫Dnxxq¯2q2xxdx+2Re∫Dnxxq¯2xq2xdx−2Re∫DẆ2E¯txxxx+γE¯xxxxdx−2Re∫DgE¯txxxx+γE¯xxxxdx+q232.


## 4. Existence and Uniqueness of Solutions

By the above a priori estimates, we prove the existence and uniqueness of solutions for stochastic Equations (7) and (8) with initial boundary conditions (9) and (10) in spaces E1.Theorem 7.
Assumen1,n0,E0∈E1, f∈L2D, q1∈H01D, and g,q2∈H2D∩H01D. Then, there is a unique solution nt,n,E∈L∞ℝ+,E1 almost surely for Equations (7) and (8). And nt,n,E is continuous from ℝ+ to E1.Proof.
First, we assumen1,n0,E0∈E1. Let eii=1∞ be the eigenvectors of Laplace operator −Δ on D with Dirichlet boundary condition, which is also an orthonormal basis of L2D. Let Pk be the projection from L2D onto the space spanned by ei:i=1,2,⋯,k. We define ntk,nk,Ek as the Galerkin approximate solution of the following equations:
(106)1λ2nttk−nxxk−PkEkxx2−αntxxk=fk+Ẇ1k,(107)iEtk+Exxk−PknkEk+iγEk=gk+Ẇ2k,t∈R+,x∈D,(108)nk0=Pkn0,ntk0=Pkn1,Ek0=PkE0.Here,fk=Pkf, gk=Pkg, Ẇ1k=PkẆ1, and Ẇ2k=PkẆ2, and Pk commutes with the operator Δ. Now, we will treat the above equations pathwise by introducing the random processes solving
(109)1λ2ηtt−ηxx−αηtxx=Ẇ1,iξt+ξxx+iγξ=Ẇ2,with Dirichlet boundary conditions
(110)η∂D=0,ξ∂D=0,and initial conditions
(111)ηx,0=0,ηtx,0=0,ξx,0=0,x∈D.Following the same method as in Section3, for any T>0 and almost all sample point ω∈Ω, we have
(112)ηt∈C0,T;H1D,η∈C0,T;H1D∩H01D,(113)ξ∈C0,T;H3D∩H01D,and satisfy the following estimates:
(114)Eηtt12+ηt22+ξt32≤C,for some positive constant C independent of T. And for any T>0,
(115)Esup0≤t≤Tηtt12+ηt22+ξt32≤CT,for some positive constant CT. Now, let Rk,M=Ntk,M,Nk,M,Fk,M be the solution of the following equations:
(116)1λ2Nttk,M−Nxxk,M−χMRk,ME1PkEkxx2−αNtxxk,M=fk,(117)iFtk,M+Fxxk,M−χMRk,ME1Pknk,MEk,M+iγFk,M=gk,with initial conditions
(118)Ntk,Mx,0=Pkn1,Nk,Mx,0=Pkn0,Fk,Mx,0=PkE0.Here,nk,M=Nk,M+Pkη and Ek,M=Fk,M+Pkξ. And χM∈C0∞ℝ such that χM=1 for r≤M and χM=0 for r≥M. Note that (116) and (117) are random differential equations with Lipschitz nonlinearity in finite dimension. Then, for almost all sample point ω∈Ω, we have a unique solution Ntk,M,Nk,M,Fk,M for (116) and (117). Define the stopping time by τM=inft>0:RE1k,M≥M, if the set RE1k,M≥M is nonempty; otherwise, τM=∞.SinceτM is increasing in M, let τ∞=limM→∞τM almost surely. And for t<τM, we have
(119)Ntk,M,Nk,M,Fk,M+Pkηt,Pkη,Pkξ,satisfying (106) and (107). By the estimates given in Section 3 and (114) and (115), for any t≥0, we have
(120)ENtk,M,Nk,M,Fk,ME12≤Cn0,n1,E0,f,g,q1,q2,with Cn0,n1,E0,f,g,q1,q2 independent of T and M, and for any T>0,
(121)Esup0≤t≤T∧τMNtk,M,Nk,M,Fk,ME12≤CTn0,n1,E0,f,g,q1,q2,with Cn0,n1,E0,f,g,q1,q2 independent of M; here, T∧τM=minT,τM. On the other hand, we have
(122)ENtk,MT∧τM,Nk,MT∧τM,Fk,MT∧τME12≥EIτM≤TNtk,MT∧τM,Nk,MT∧τM,Fk,MT∧τME12≥M2ℙτM≤T,where IτM≤T=1 for τM≤T and IτM≤T=0 for τM>T. Then, according to (120), we have
(123)ℙτM≤T≤1M2Cn0,n1,E0,f,g,q1,q2.According to the above estimate and Borel-Cantelli lemma, for anyT>0, we have ℙτM>T=1. So we know
(124)Ntk,Nk,Fk=limM→∞Ntk,M,Nk,M,Fk,M,satisfying the following random differential equations:
(125)1λ2Nttk−Nxxk−PkEkxx2−αNtxxk=fk,(126)iFtk+Fxxk−PknkEk+iγFk=gk,with initial conditions
(127)Nk0=Pkn0,Ntk0=Pkn1,Fk0=PkE0.Then,Ntk,Nk,Fk satisfies the estimates (120) and (121), and for any t≥0,
(128)ntkt,nkt,Ekt=Ntkt,Nkt,Fkt+Pkηtt,Pkηt,Pkξkt,is the unique global solution of (106), (107), and (108). Now, we will consider (125) and (126) for fixed ω. First, by (121), for any T>0,
(129)ℙ⋂L=1∞⋃l=1∞⋂k=l∞sup0≤t≤TNtkt,Nkt,FktE12≥L=0.Let(130)Ω~=⋂L=1∞⋃l=1∞⋂k=l∞sup0≤t≤TNtkt,Nkt,FktE12≤L.Then,ℙΩΩ~=0. Now, for any fixed ω∈Ω~, there is an rω with 0<rω<∞ such that
(131)sup0≤t≤TNtkt,Nkt,FktE12≤rω.Then, we can extract a subsequence still denoted byNtk,Nk,Fk such that for any T>0, Ntk converges to Nt weakly star in L∞0,T;L2D, Nk converges to N weakly star in L∞0,T;H01D, and Fk converges to F weakly star in L∞0,T;H2∩H01D.These convergences are sufficient to pass the limitk→∞ in linear terms, but we need a strong convergence of Fk for nonlinear terms. In fact from (126) and estimate (131), we know Fk is bounded in L∞0,T;L2D. Then, by Lemma 1, we can further extract a subsequence still denoted by Fk such that Fk converges to F strongly in L20,T;H01D. Then, by a standard procedure, we can pass the limit k→∞ to show that Nt,N,F∈L∞0,T;E1 is a weak solution of
(132)1λ2Ntt−Nxx−Exx2−αNtxx=f,iFt+Fxx−nE+iγF=g,with initial conditions
(133)Ntx,0=n1x,Nx,0=n0x,Fx,0=E0x,x∈D.Then,nt,n,F=Nt,N,F+ηt,η,ξ is a solution of (7) and (8) and satisfies the estimates in Section 3.Now, we prove the continuity of the solution. In fact, forω∈Ω~, f∈L2D, and Exx2∈L∞0,T;L02D, we have
(134)1λ2Ntt−Nxx∈L∞0,T;L02D.Then, by Lemma 4.1 and Chapter II of [23], we have for almost all ω∈Ω
(135)N∈C0,T;H01D,Nt∈C0,T;L2D.By a similar method, noticingg∈H01D and Ft∈L∞0,T;L02D almost surely, by [21], we have
(136)F∈C0,T;H02D⋂H01D.Then, by (113) and the definition of N and F,
(137)nt,n,F∈C0,T;E1,almost surely. Now, as the noise is additive, we can follow the same approach as in [11]. The solution nt,n,E is unique in L∞0,T;E1 almost surely and is continuous from 0,T to E1 almost surely. Then, for T is arbitrary, by the estimates in Section 3, we derive Theorem 7.Using the same method as above, we obtain the following results.Theorem 8.
Assumen1,n0,E0∈E2, f∈H01D, g,q1∈H2D∩H01D, and q2∈H3D∩H01D. Then, there is a unique solution nt,n,E∈L∞ℝ+,E1 almost surely for Equations (7) and (8). And nt,n,E is continuous from ℝ+ to E1.


## 5. Conclusion

In [10], the authors studied the initial boundary value problem for a generalized Zakharov system. The authors proved the global existence and uniqueness of the generalized solution to the problem by a priori estimates and Galerkin method. In this paper, we discuss the random forced strongly dissipative Zakharov equations, which are the generalized Zakharov system under random influences in [10]. We proved the existence and uniqueness of solutions in energy spaces E1 and E2. The results of this paper are a good supplement to the results in [10].



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