Global Existence for Stochastic Strongly Dissipative Zakharov Equations

where the wave fields Eðx, tÞ and nðx, tÞ 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:


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 ℝ d+1 is given by where the wave fields Eðx, tÞ and nðx, tÞ 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][3][4][5][6][7][8][9][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: 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][12][13]). In [14], the authors considered the following strongly dissipative Zakharov equations: 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][16][17][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 domain D in the Itô sense: where the parameters γ > 0 and α > 0, and W 1 and W 2 are independent L 2 ðDÞ-valued Wiener processes, which will be detailed in the next section. Here, _ W denotes the general derivative of the Wiener processes W with respect to time.
The rest of this paper is organized as follows. In Section 2, 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.

Preliminary
In this section, we give a detailed description of the stochastic strongly dissipative Zakharov equations. Let D = ½0, l with 0 < l < ∞. Consider the following strongly dissipative stochastic Zakharov equations on a regular domain D: with initial conditions and Dirichlet boundary conditions 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 = n t + εn, where ε > 0 is small enough and will be chosen later. So we have the following equations: 1 Here, we give a complete probability space ðΩ, F, fF t g t≥0 , ℙÞ: The expectation operator with respect to ℙ is denoted by E: The stochastic terms W 1 ðtÞ and W 2 ðtÞ on ðΩ, F, fF t g t≥0 , ℙÞ are defined by where ω 1 ðtÞ is a standard real-valued Wiener process and ω 2 ðtÞ is a standard complex-valued Wiener process independent of ω 1 ðtÞ: In addition, q 1 ðxÞ and q 2 ðxÞ are sufficiently smooth functions. We will work on the usual functional spaces L 2 ðDÞ, H m ðDÞ, and H 1 0 ðDÞ: The inner product on L 2 ðDÞ will be denoted by ðu, vÞ = Re Ð D uðxÞ vðxÞdx, u, v ∈ L 2 ðDÞ and the norm by In our approach, we need the following lemmas. Let ðX, k·k X Þ ⊂ ðY, k·k Y Þ ⊂ ðZ, k·k Z Þ be three reflective Banach spaces and X ⊂ Y with compact and dense embedding. Define the Banach space Advances in Mathematical Physics Then, we have the following lemma about compactness result (see [21]).
Another lemma is needed for some maximal estimates on the stochastic integral. Assume U and H are separable Hilbert spaces, W is a Q-Wiener process on U 0 with U 0 = Q 1/2 U: And let L 0 2 = L 0 2 ðU 0 , HÞ be the space of Hilbert-Schmidt operators from U 0 to H: We have the following results (see [22]).

Lemma 2.
For any r ≥ 1 and any L 0 2 -valued predictable process where c r and C r are some positive constants dependent on r.

Time Uniform A Priori Estimates
In this section, in the sense of expectation, we give a priori estimates for ðn t , n, EÞ in different spaces E 0 , E 1 , and E 2 .
Taking the supremum and expectation on both sides of (21), by Lemma 2 and (23), for any T > 0, there exists a positive constant C T ; we have where C T ðE 0 , g, q 2 Þ is a constant depending on kE 0 k 0 , kgk 0 , kq 2 k 0 and T. By (24), we obtain E ∈ L 2 ðΩ ; L ∞ ð0, T ; L 2 ðDÞÞÞ: By the above estimates, we can further give an estimate of kEk 2p 0 for any p ≥ 1. Now applying Itô's formula, Young inequality, and Hölder inequality, we have 3 Advances in Mathematical Physics Integrating (25) from 0 to t, we obtain Taking expectation on both sides of (26), we get Then, by Gronwall inequality, we get where Cðg, q 2 Þ is independent of T. By (28), we obtain E ∈ L ∞ ð0,∞;L 2p ðΩ, L 2 ðDÞÞÞ.
Taking the supremum and expectation on both sides of (26), by Lemma 2 and (28), for any T > 0, there exists a positive constant C T ; we have where C T ðE 0 , g, q 2 Þ is a constant depending on kE 0 k 0 , kgk 0 , kq 2 k 0 and T. By (29), we obtain E ∈ L 2p ðΩ ; L ∞ ð0, T ; L 2 ðDÞÞÞ. Lemma 3 is proved completely.
On the other hand, integrating from 0 to t on both sides of (47), deduce Now, taking the supremum and expectation on both sides of the above inequality, we have Then, by (28), (54), and (56), we obtain where C T depends on the initial data. Therefore, by (48), we have By (58), we obtain E ∈ L 2 ðΩ ; L ∞ ð0, T ; E 0 ÞÞ.

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In addition, from (52), we have Integrating from 0 to t and taking the supremum and the expectation on both sides of (59), we have By (50), (51), Hölder inequality, and Young inequality, we have the following estimates: By (28), (54), (60), and (61), we have Moreover, by (29), (51), and (62), we have where C T depends on the initial data.

First as
By (81) and (84), we have Applying Itô's formula to H t ð Þ Taking expectation on both sides of the above inequality, by Hölder inequality and (88), we derive By Gronwall inequality, we have where CðE 0 , f , g, q 1 , q 2 , n 0 , n 1 Þ is independent of T.
By the same approach of Lemma 4, we obtain where C T depends on the initial data. Therefore, by (47), we have By (92), we obtain E ∈ L 2 ðΩ ; L ∞ ð0, T ; E 1 ÞÞ. Moreover, we get By (87), we have where C T depends on the initial data. Therefore, by (94), we have By (95), we obtain E ∈ L 2p ðΩ ; L ∞ ð0, T ; E 1 ÞÞ. Lemma 5 is proved completely.

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 E 1 .
Proof. First, we assume ðn 1 , n 0 , E 0 Þ ∈ E 1 . Let fe i g ∞ i=1 be the eigenvectors of Laplace operator −Δ on D with Dirichlet boundary condition, which is also an orthonormal basis of L 2 ðDÞ: Let P k be the projection from L 2 ðDÞ onto the space spanned by fe i : i = 1, 2, ⋯, kg: We define ðn k t , n k , E k Þ as the Galerkin approximate solution of the following equations: with initial conditions Here, n k,M = N k,M + P k η and E k,M = F k,M + P k ξ. And χ M ∈ C ∞ 0 ðℝÞ such that χ M = 1 for jrj ≤ M and χ M = 0 for jrj ≥ 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 ðN k,M t , N k,M , F k,M Þ for (116) and (117). Define the stopping time by satisfying (106) and (107). By the estimates given in Section 3 and (114) and (115), for any t ≥ 0, we have with Cðn 0 , n 1 , E 0 , f , g, q 1 , q 2 Þ independent of T and M, and for any T > 0, with Cðn 0 , n 1 , E 0 , f , g, q 1 , q 2 Þ independent of M; here, T ∧ τ M = min fT, τ M g. On the other hand, we have where Iðτ M ≤ TÞ = 1 for τ M ≤ T and Iðτ M ≤ TÞ = 0 for τ M > T. Then, according to (120), we have According to the above estimate and Borel-Cantelli lemma, for any T > 0, we have ℙðτ M > TÞ = 1. So we know satisfying the following random differential equations: iF k t + F k xx − P k n k E k 17 Advances in Mathematical Physics with initial conditions N k 0 ð Þ = P k n 0 , N k t 0 ð Þ = P k n 1 , F k 0 ð Þ = P k E 0 : Then, ðN k t , N k , F k Þ satisfies the estimates (120) and (121), and for any t ≥ 0, 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, Then, ℙðΩΩÞ = 0. Now, for any fixed ω ∈Ω, there is an rðωÞ with 0 < rðωÞ<∞ such that Then, we can extract a subsequence still denoted by ðN k t , N k , F k Þ such that for any T > 0, N k t converges to N t weakly star in L ∞ ð0, T ; L 2 ðDÞÞ, N k converges to N weakly star in L ∞ ð0, T ; H 1 0 ðDÞÞ, and F k converges to F weakly star in L ∞ ð0, T ; H 2 ∩ H 1 0 ðDÞÞ. These convergences are sufficient to pass the limit k → ∞ in linear terms, but we need a strong convergence of F k for nonlinear terms. In fact from (126) and estimate (131), we know F k is bounded in L ∞ ð0, T ; L 2 ðDÞÞ: Then, by Lemma 1, we can further extract a subsequence still denoted by F k such that F k converges to F strongly in L 2 ð0, T ; H 1 0 ðDÞÞ. Then, by a standard procedure, we can pass the limit k → ∞ to show that ðN t , N, FÞ ∈ L ∞ ð0, T ; E 1 Þ is a weak solution of with initial conditions Then, ðn t , n, FÞ = ðN t , N, FÞ + ðη t , η, ξÞ is a solution of (7) and (8) and satisfies the estimates in Section 3. Now, we prove the continuity of the solution. In fact, for ω ∈Ω, f ∈ L 2 ðDÞ, and jEj 2 xx ∈ L ∞ ð0, T ; L 2 0 ðDÞÞ, we have Then, by Lemma 4.1 and Chapter II of [23], we have for almost all ω ∈ Ω By a similar method, noticing g ∈ H 1 0 ðDÞ and F t ∈ L ∞ ð0, T ; L 2 0 ðDÞÞ almost surely, by [21], we have Then, by (113) and the definition of N and F, almost surely. Now, as the noise is additive, we can follow the same approach as in [11]. The solution ðn t , n, EÞ is unique in L ∞ ð0, T ; E 1 Þ almost surely and is continuous from ½0, T to E 1 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.

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 E 1 and E 2 . The results of this paper are a good supplement to the results in [10]. 18 Advances in Mathematical Physics