Research Article | Open Access
Global Existence for Stochastic Strongly Dissipative Zakharov Equations
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 spaces and , by using the standard Galerkin approximation method of stochastic partial differential equations.
The Zakharov system, derived by Zakharov in 1972 , describes the interaction between Langmuir (dispersive) and ion acoustic (approximately nondispersive) waves in an unmagnetized plasma. The usual Zakharov system defined in the space is given by where the wave fields and 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 , 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 , the author studied the following dissipative Zakharov equations: where positive constants and are damping coefficients and and 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 , the authors considered the following strongly dissipative Zakharov equations:
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 , stochastic dissipative Zakharov equations have been studied on a regular domain 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 , 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 in the Itô sense: where the parameters and , and and are independent -valued Wiener processes, which will be detailed in the next section. Here, denotes the general derivative of the Wiener processes 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 throughout this paper. Now, we state the main results of the paper.
In this section, we give a detailed description of the stochastic strongly dissipative Zakharov equations. Let with . Consider the following strongly dissipative stochastic Zakharov equations on a regular domain : with initial conditions and Dirichlet boundary conditions here, , , , , and are given, with and .
As in , to study the solution of strongly dissipative stochastic Zakharov equations, we set , where is small enough and will be chosen later. So we have the following equations:
Here, we give a complete probability space The expectation operator with respect to is denoted by The stochastic terms and on are defined by where is a standard real-valued Wiener process and is a standard complex-valued Wiener process independent of In addition, and are sufficiently smooth functions.
We will work on the usual functional spaces , , and The inner product on will be denoted by and the norm by . The general -norm of is denoted by And the norm on is denoted by . The real and imaginary parts of a complex number are denoted, respectively, by and .
Now, we define spaces as , , and . Endow with the usual product norm . Then, with compact embeddings.
In our approach, we need the following lemmas. Let be three reflective Banach spaces and with compact and dense embedding. Define the Banach space with
Then, we have the following lemma about compactness result (see ).
Lemma 1. If is bounded in , then is precompact in .
Another lemma is needed for some maximal estimates on the stochastic integral. Assume and are separable Hilbert spaces, is a -Wiener process on with And let be the space of Hilbert-Schmidt operators from to We have the following results (see ).
Lemma 2. For any and any -valued predictable process we have where and are some positive constants dependent on .
3. Time Uniform A Priori Estimates
In this section, in the sense of expectation, we give a priori estimates for in different spaces , , and .
Lemma 3. Assume Then, for any and ,
Proof. Applying Itô’s formula to , by (13), we have
Integrating (20) from 0 to , by Young inequality, we obtain
Then, by Gronwall inequality, we get where is independent of .
By (23), we obtain .
By (24), we obtain
By the above estimates, we can further give an estimate of for any . Now applying Itô’s formula, Young inequality, and Hölder inequality, we have
Integrating (25) from 0 to , we obtain
Taking expectation on both sides of (26), we get
Then, by Gronwall inequality, we get where is independent of .
By (28), we obtain .
By (29), we obtain .
Lemma 3 is proved completely.
Lemma 4. Assume and . Then, for any and , .
Proof. Applying Itô’s formula to , by (12), we have
Since , from (30), we obtain where is the first eigenvalue of .
By (31), we get
Applying Itô’s formula to , by (13), we have
Substituting into (33), we have
In addition, applying Itô’s formula to , we have
By Hölder inequality and Young inequality, we have the following estimates:
By (38), we obtain
By (39), we get
Now, taking and letting , the above inequality is changed into
Integrating (41) from 0 to , we obtain
Taking expectation on both sides of the above inequality, by (28), we get
By (23) and Gronwall inequality, we have where is independent of .
Since we have