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Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises
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.
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 . Peszat and Zabczyk  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 , 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 , 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 -smooth domain with sufficient smooth boundary, we consider the following stochastic Boussinesq equations driven by subordinator Lévy noise: where is the velocity vector, is salinity, is the pressure term, , denotes the Laplacian operator, and denotes the gradient operator. Moreover, is the Froude number, is the Reynolds number, is the Prandtl number, and is a unit vector in the upward vertical direction. The initial data , are given. Both and are subordinator Lévy processes on Hilbert spaces and , 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 Section 2, we first present some properties of the subordinator Lévy process , then review some fundamental properties of the stochastic integral with respect to Lévy process . 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.
In this section, we introduce some operators and fraction spaces and then present some properties of the subordinator Lévy process and the stochastic integral with respect to Lévy process .
In order to reformulate the stochastic Boussinesq equations (1) as an abstract stochastic evolution, we introduce the following function spaces.
Denote to be the space of functions defined on , which are -integrable with respect to the Lebesgue measure , endowed with the usual scalar product and norm, that is, for ,
For and , define as the usual Soblev space with scalar product and the induced norm where is the th order weak derivative of .
For , let be defined by the complex interpolation method  as follows. where , , and are chosen to satisfy The closure of in the Banach space , , , will be denoted by .
The following product spaces are needed:
Let denote the closure of with respect to the -norm, denote the closure of with respect to the -norm, and be the dual space of . In particular, we denote by and and , respectively.
Denote where and .
Now, we define the following two operators:
Then, the stochastic Boussinesq system (1) can be rewritten as the following abstract stochastic evolution equation:
In order to apply the technique of the reproducing Kernel Hilbert space, it is better to introduce the definition -radonifying.
Definition 1 (see ). Let and be Banach spaces, a bounded linear operator is called -radonifying if and only if is -additive, where is the canonical cylindrical finitely additive set-valued function (also called a Gaussian distribution) on .
The following is our standing assumption:
Assumption 1. Space is a Hilbert space such that for some ,
Remark 2. Under the above assumption, we have the facts that and the Banach space is taken as (see [11, 14, 15] for more details and related results). In fact, space is the reproducing kernel Hilbert space of noise on .
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 ). We will present some properties of the subordinator Lévy process , , then review briefly the stochastic integral with respect to Lévy process .
Definition 3 (see [1, 2, 11]). Let be a Banach space, and let be an -valued stochastic process defined on a probability space . Stochastic process is called a Lévy process if (L1), a.s.;(L2) process has independent and stationary increments; and(L3) process is stochastically continuous, that is, for all and for all ,
A subordinator Lévy process is an increasing one-dimensional Lévy process.
For , denotes the set of all subordinator Lévy processes , whose intensity measure satisfies the condition .
In the most interesting cases, the space is a subspace of , that is, , and where is an independent subordinator process belonging to class , , is an -valued cylindrical Wiener process defined on some Banach space .
We decompose the -valued Lévy process into two parts and , the first one with small jumps and the second one with (relatively) large jumps, that is, with being the intensity measure of Lévy process , being the Lévy process with the intensity measure: and be the Lévy process with the intensity measure . Then can be defined as a compound Poisson process with the intensity measure , and , can be defined by the Poisson random measure which is defined as follows: where , . Here, the symbol denotes the increment of .
We assume that the process is right-continuous with left-hand side limits. Thus
Notice that as is a time homogenous Poisson random measure, can be expressed as Hence,
Assume that the operator , , is a strongly measurable function taking values in the space of all bounded linear operator from to . Let be the jump times for and , . Then, the stochastic integral of with respect to jump process , , can be defined as
Since the operator is taking values in , it follows from the decomposition of that the sum of sequences is finite. Hence the stochastic integral of the operator with respect to is taking values in . Moreover, the stochastic integral of the operator , , with respect to Lévy process can be defined by and takes values in as well (see  for more details).
Next, we recall some basic definitions and properties for general random dynamical systems, which are taken from . Let be a complete separable metric space and be a probability space.
Definition 4. is called a metric dynamical system if the mapping is measurable, , for all , and for all .
Definition 5. A random dynamical system (RDS) with time on over on is a -measurable map: such that (i) (identity on ) for any ,(ii) (Cocycle property) for all and .
An RDS is said to be continuous or differentiable if for all , and an arbitrary outside outside -nullset , the map is continuous or differentiable, respectively.
Assume that the bounded linear operator generates a -semigroup on a Hilbert space and defined on a filtered probability space is a subordinator Lévy process taking values in a Hilbert space .
Consider the following stochastic Langevin equation:
Definition 6. Let be a square integrable -measurable random variable in . A predicable process is called a mild solution of the Langevin equation (24) with initial data if it is an adapted -valued stochastic process and satisfies
It is well known that the Ornstein-Uhlenbeck process , , has some important integrability. Here we need the Banach space to be of type , for . First we recall the definition briefly (see  for more details).
Definition 7 (see ). For , the Banach space is called as type , if and only if there exists a constant such that for any finite sequence of symmetric independent identically distribution random variables , , and any finite sequence from , satisfying
Moreover, if there exists a constant such that for every -valued martingale , , satisfying the separable Banach space is called a separable martingale type -Banach space.
Lemma 8 (see [11, Corollary 8.1, Proposition 8.4]). Assume that , is a subordinator Lévy process from the class , is a separable type -Banach, is a separable Hilbert space , , and is a -valued Wiener process.
Define the -valued Lévy process as and define the process as Then, with probability , for all ,
Theorem 9. Assume that , , is the semigroup generated by the bounded linear operator in the space . Then, if one of the following conditions is satisfied: (i) or (ii) and the Banach space is of separable martingale type -Banach space, the Langevin equation (24) admits one mild solution , . Moreover, if , , , is a -group in the separable martingale type -Banach space , then the mild solution of the Langevin equation is a cádlág (right-continuous with left-hand side limits) process.
Proof. As , , is a -group in the separable martingale type -Banach space , the Hilbert space is the reproducing kernel Hilbert space of , and the embedding operator satisfies the -radonifying property. The proof of Theorem 9 is just a simple application of Theorems 4.1 and 4.4 in .
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 both and are positive definite, self-adjoint operators, and denote and to be the domains of and , respectively. Hence, the domain of the operator can be represented as . It follows from Lemma 2.2 in  that there exists positive numbers , such that Let . Then
For any arbitrary , we can define the following trilinear form by We have the following results.
Lemma 10 (see [7, Lemma 2.3]). If , then
Lemma 11 (see [7, Lemma 2.4]). There exists a constant such that if , , , then(1), , , ,(2), , , ,(3), , ,(4), , , ,(5), , , .
Definition 12. An -valued adapted and -valued cádlág process is considered as a solution to (11), if for each , and for any , and for any , -a.s.,
Lemma 13. Assume that , , and . Then there exists a unique such that Moreover, where and the mapping is analytic.
Proof. It can be shown by the same approach as the one in Proposition 8.7 in .
Lemma 14 (see [2, Proposition 10.1]). Let be a continuous function whose left derivative exists at . Then the function , , is left differentiable at and
Definition 15. Let be a separable Banach space, be the dual space of . The subdifferential of norm at is defined by the formula
A mapping is said to be dissipative, if for any , there exists such that
A dissipative mapping is called an -dissipative mapping or maximal dissipative if the image of is equal to the whole space for some (and then for any ), that is,
Assume that is an -dissipative mapping. Then its resolvent and respectively the Yosida approximations , , are defined by
Lemma 16 (see [2, Proposition 10.2]). Let be an -dissipative mapping on . Then (1) for all and , ;(2) the mapping , , are dissipative and Lipschitz continuous: Moreover, , for all ; and(3), for all .
The following theorem is one of the main results of this paper, which will be proved by applying the well-known Yosida approach.
Proof. Denote to be the stationary solution of Langevin equation (24). Let . Then (11) is converted into the following evolution equation with random coefficients:
where generates an analytic -semigroup (see Section 2.2 in ). It follows from the proof of Theorem 10.1 in  that, for , , and sufficiently small , the mappings and are -dissipative. Hence, the Yosida approximations of the -dissipative mappings and can be respectively denoted by
Now consider the following random approximate equation: It is easy to verify that generates an analytic -semigroup . Notice that the Yosida approximate operators are Lipschitz. Therefore the random approximation equation (50) has a unique continuous solution .
Next we will show that in , and this limit is actually the mild solution of stochastic Boussinesq equation (48).
For the sake of simplification, we just present the estimations when , and the remaining estimates can be obtained by the similar arguments for .
Let be the solution of the integral equation: Notice that the operator is Lipschitz continuous and is cádlág. Hence, there exists a solution of random approximate equation (50), which is continuous in .
For and , direct computation implies Since both and are -dissipative. Therefore, there exists constant , , and such that for all , , Then By the Hille-Yosida theorem, it follows that uniformly in on compact subsets of .
Hence, it follows that uniformly on bounded intervals as .
By Gronwall inequality, we have
By Lemma 14, Recalling that both and are -dissipative and is linear, we obtain that is, It follows from the estimate (58) that, for any , and ,
Similarly, by Lemma 16, for , By the dissipation of the operators , , and and estimates (63), there exists a constant such that Therefore By the estimate (58), Thus, in uniformly on as .
Next, we are going to show that the solution of the Yosida approximations equation is a mild solution: By the reflexivity of and the estimate , , , there exists a subsequence , which converges weakly in and weakly converges to the function in . Since is strong convergent in , and Let , then Moreover Notice that weakly converges in . So, letting , we obtain It follows from the arbitrariness of that Thus, is a mild solution of random Boussinesq equation (50).
Theorem 18. For any , the map defined by the solution of stochastic Boussinesq equation (11) as has the cocycle property; that is, the solution of stochastic Boussinesq equation (11) generates a random dynamical system .
Proof. From Theorem 17, stochastic Boussinesq equation (11) admits a unique solution . Define the map
(i) By the similar argument of Theorem 17, every solution of the Yosida approximation equation (50) is measurable. Notice that uniformly as . Hence, the limit function is also measurable. Thus, the mapping is measurable.(ii) Obviously, .(iii) It suffices to verify that the cocycle property holds for the mapping , that is,
In fact, recalling that , it follows that
Therefore, we obtain
The uniqueness of the solution implies that almost surely holds, that is,
Thus, the cocycle property for the mapping holds.
By the definition of random dynamical systems , the solution mapping of the stochastic Boussinesq equation (11) generates a random dynamical system . Thus, the proof of Theorem 18 is complete.
In Section 3, 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 , we refer the similar argument to .
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 where and are -valued Brownian motion, and are constants vector in , and are measurable mappings from some measurable space to , and and are compensated Poisson measure on with intensity measure and , respectively, where and are -finite measure on , , and satisfying
Let be the space of all cádlág paths from to endowed with the uniform convergence topology. Since there are finite jumps when the character measure , we can rearrange the jump time of as . Since there is no jump on the interval , just as the approach in , we can apply Banach fixed point theorem to prove that there exists a unique solution in . Define On , define Similar to the argument in , since is stationary Poisson point process on with intensity measure , then is also a stable Poisson point process on with intensity measure . Define