Research Article | Open Access
-Random Attractors and Asymptotic Smoothing Effect of Solutions for Stochastic Boussinesq Equations with Fluctuating Dynamical Boundary Conditions
This work is concerned with the random dynamics of two-dimensional stochastic Boussinesq system with dynamical boundary condition. The white noises affect the system through a dynamical boundary condition. Using a method based on the theory of omega-limit compactness of a random dynamical system, we prove that the -random attractor for the generated random dynamical system is exactly the -random attractor. This improves a recent conclusion derived by Brune et al. on the existence of the -random attractor for the same system.
The Boussinesq equations are a coupled system of the Navier-Stokes equations and the scalar transport equation for fluid salinity, temperature, or density. These Boussinesq equations models various phenomena in environmental, geophysical and climate system, for example, oceanic density currents and the thermohaline circulation; see, for example, [1–3]. In this paper, the scalar quantity in the considered Boussinesq system is salinity, with dynamical boundary condition for the salinity.
Letbe a bounded domain with the-smooth boundary, in the vertical plane. Let,, andbe independent two-sided real-valued Wiener processes with values in appropriate function spaces. This paper is mainly concerned with the long time behavior of the solutions to the following Boussinesq equations with general additive noises and fluctuating dynamical boundary conditions : with velocity, salinity, and pressure. In (1),is the Froude number;is the Reynolds number; andis the Prandtl number.is the Laplacian operator;is the gradient operator; div is the divergence operator;is the trace operator with respect to the boundary.is a unit vector in the upward vertical direction (opposite to the gravity). Finally,,is a given function describing the mean salinity flux through the boundary;andare the initial conditions;is the outer normal derivative. Without loss of generality, in this paper we take,,,andto be 1.
Qualitatively, the Boussinesq system (1) emphasizes the random dynamical boundary condition which modes the interaction of boundary and the domain. In particular, ifdescribes the temperature then the heat exchange between physical domain and its boundary can be modeled; see, for example, . For this Boussinesq system (1), the large deviation principle via a weak convergence approach is studied in , recently. In , they derived a priori estimates for the existence of random absorbing sets and showed that the random dynamical system (RDS) generated by the solution of (1) had a random attractor inspace. When the random dynamical boundary condition in (1) is replaced by the nonhomogeneous boundary condition, [6, 7] proved the existences of random attractors for this system with multiplicative noise and additive noises inspace, respectively. For the study of the random attractor about other models integrated the Navier-Stokes equations, we refer to  for the MHD equations.
In recent years, the theory of random attractors for some concrete dissipative stochastic partial differential equations has been studied by many authors; see, for example, [6, 8–12]; ever since [13, 14] launched their fundamental work on the RDS. Such an attractor, which generalizes nontrivially the global attractors well developed in [15–18] and so forth, is a compact invariant random set which attracts every orbit in the phase space. It is uniquely determined by attracting deterministic compact sets of phase space . In order to obtain the existence of random attractors, one always need to show the existence of a compact random absorbing set in the sense of absorption . This can be achieved by employing the standard Sobolev compact embedding of several functional spaces, when the generated RDSs are defined in some bounded domains; see, for example, [6–10, 20] and references cited there.
In this paper, we are interested in the existence of random attractors of the Boussinesq system (1) inspace which is stronger thanspace. It is pointed out that if the initial data belongs to, the solution to the system (1) enters into the spaceand has no higher regularity, see . Hence the Sobolev compact embedding cannot be employed in. Here, we try to overcome the obstacle of compact imbedding by using the notion of omega-limits compactness, which was initiated in [12, 21] in the framework of RDS. This type of compactness is equivalent to the asymptotic compactness [22, 23] in some spaces and can be proved by check the flattening condition; see . More precisely, we prove the existence of the-random attractor for this RDS (for the bispaces random attractors; the reader is referred to [11, 16–18]). To this end, a so qualitatively new method is necessary, for instance, to derive a priori estimates of the solutions such that the flattening condition holds inspace and then the necessary omega-limits compactness for the RDS inspace is followed; see Lemma 12 in Section 4. The main advantage of this technique is that we need not to estimate the solutions in functional spaces of higher regularity to demonstrate the existence of compact random absorbing set which does not work in this case. This method has been used recently to obtain the existence of random attractors inspace for the stochastic reaction-diffusion equations [24, 25].
The conclusion in this study shows that the-random attractor is qualitatively an-random attractor. This implies that the solutions to (1) become eventually more smoothing than the initial data.
The outline of this paper is as follows. Section 2 presents some functional settings for the Boussinesq system. Section 3 lists the conditions and the main conclusion of this note. In Section 4, we prove some estimates for the solution orbits inandand then prove our main conclusion.
2. Functional Settings
We recall some function spaces and operators that we will be used in the following discussion.
Let endowed with the scalar inner productand with norm denoted by. This notation is also used to denote the norm inandwithout any confusion.
We define a functional spaceintegrated the boundary and also the divergence free condition:
Letthe closure ofinandbe the closure ofin, and the norm inbeing denoted by . From the above argument, the spaceis equipped with the norm: which is equivalent to the natural norm:.
For,, we define the operators
By Lemma 2.2 in , the operatoris a positive self-adjoint unbounded operator and has the Poincaré inequality: Thenis also self-adjoint but compact operator in, so we can utilize the elementary spectral theory in a Hilbert space. We infer that there exists a compete orthonormal family in, also in,of eigenvectors of. The corresponding spectrum ofis discrete and denoted bywhich are positive, increasing, and tend to infinity as.
In particular, we also can use the spectrum theory to allow us to define the operator, the power of. For, the operatoris also an strictly positive and self-adjoint unbounded operator inwith a dense domain. This allows us to introduce the function spaces, This normonis equivalent to the usual norm induced by; see Temam  for details. In particular,and.
Based on the orthonormal basisof eigenfunctions of, we define the-dimensional subspaceand the canonical orthogonal projectionsuch that for every,has a unique decomposition:, where that is,.
When the projectionoperates on the first component of, one can easily show that for, the nature numbers set.
We also state the well-known Brezis-Gallouet’s inequality with Dirichlet boundary condition in two dimension case; see  or Proposition 3 in ; there exists a positive constantsuch that whereis the first eigenvalue of the Stokes operator. By (12) we have
According to the above notations, we can write (1) as the following abstract evolution equation form: where.
3. Existence of Random Attractor in
In order to model the noise in the initial problem (1), we need to define a metric dynamical system (MDS) which is a group of measure preserving transformations on a probability space. For the definition of the MDS we refer to [28, 29] and so forth.
A standard model for a spatially correlated noise is the generalized time derivative of a two-sidedBrownian motion,. Letbe the separable Hilbert space defined in Section 2. As usual, we introduce the spatially valued Brownian motion MDS , where with compact open topology. This topology is metrizable by the complete metric: whereforandin.is the Borel--algebra induced by the compact open topology of. Suppose the Wiener processhas covariance operator. Letbe the Wiener measure with respect to. The Wiener shift is defined by Then the measureis ergodic and invariant with respect to the shift.
The associated probability space defines a canonical Wiener process. We also note that such a Wiener processgenerates a filtration:
We introduce the following stochastic partial differential equation on: Becauseis a positive and self-adjoint operator, there exists a mild solution to this stochastic equation with the form which is called an Ornstein-Uhlenbeck process; see . For the Ornstein-Uhlenbeck process we have the regularity hypothesis; see also .
Lemma 1. Suppose that the covariance operatorof the Wiener processhas a finite trace; that is,satisfies that for someand some (arbitrary small), wheredenotes the trace of the covariance operator. Then an-measurable Gaussian variableexists, and the process is a continuous stationary solution to the stochastic equation (19). Furthermore, the random variableis tempered and the expectation
Here we denote the solution to (23) byor more briefly. By Lemma 4.7 in , the solution of the evolution equation (23) generates a continuous measurable RDSingiven by . Put. Then or brieflyis a solution to (23) with initial value.
The main conclusion of this study states the following.
Theorem 2. One supposes that (21) holds. Set that whereis a generic constant depending on the data of the problem and somethat has to be chosen sufficiently small andis the same as in (9). Assume that the mathematical expectation of Then the RDSgenerated by (1) admits a unique random attractorinin the sense that for-a.s., where denotes the Hausdorff semiistance inandis the collection of tempered subsets ofas in . Furthermore,is identical with the random attractorin.
By the well-known abstract result of Theorem 3.5 in , the existence of a random attractorin the following sense that has been obtained in . However, Our Theorem 2 shows the existence of random attractor which is compact in the space, which is stronger than. Thus the Sobolev compact embedding theorem is not available and it seems impossible to obtain a compact random absorbing set in the spacewhen the initial data belongs to. Then Theorem 3.5 in  is unapplicable to the proof of our Theorem 2.
Fortunately, Theorem 2 can be proved by checking the omega-limit compactness in the space, which is based on the viewpoint of Kuratowski measure of noncompactness. This kind of compactness can be easily obtained by showing the flattening condition, see .
For clarification, we state some general concepts used in the sequel. Letbe a Banach space with norm andbe the collection of all random subsets of.
Definition 3 (see ). An RDSonover an MDS is said to be omega-limit compact if for everyand, there existssuch that for all, whereis the Kuratowski measure of noncompactness ofdefined by
Definition 4 (see ). An RDSonover an MDS is said to possess the flattening condition if for-a.s.and every, there existand a finite dimensional spaceofsuch that for a bounded projector,
Definition 5 (see ). (i) A random variableover an MDS is tempered if for-a.s.,
(ii) A random setis called tempered ifis a tempered random variable.
Definition 6. A random setis an-random absorbing set for RDSover an MDS if for-a.s.and every, there existssuch that for all, where.
For our problem,andis the collection of tempered subsets of. The finite dimensional space and the bounded operatorfor a sufficient large, where andare defined in Section 2.
4. The Proofs of Main Result
Lemma 7. Assume that,, andare three locally integrable functions and,nonnegative onsuch that for, Then for everyand any positive constant, one has
Proof. Let. We multiply (36) byand the resulting inequality reads Hence, by integration fromto, Integrating the above inequality with respect tobetweenand, we get the desired inequality.
Lemma 8. There exist positive constantsandsuch that the followings hold: where
Proof. The inequality (40) is the same as the inequalityat page 1112 in . The inequality (41) is a combination of the formulasandat pages 1110-1111 in  with a tiny modification. Following a same calculation as page 1114 in  we can obtain (42).
Lemma 9. Assume that. Let. Then for-a.s., there exist random radiiandand constantsuch that for all, the solution of problem (23) withsatisfies that for every: whereis the collection of tempered random subsets of.
Proof. We replacewith, andwithin (40) to produce that for,
By noting that for,, then we find that for,
from which and (47) it follows that for every,
By the Birkhoff’s ergodic theorem and along with our assumption (26), it yields that
and then it implies that
Asis subexponential growth, so
Note thatis also tempered random variable andwith, whence there exists constantsuch that for allwith,
Furthermore,is tempered; see . This completes the proof of (45).
Next, we show (46). Integrating (41) fromtowith, whereis in (53), we get Then by (45) we get that for all, which gives an expression for.
Lemma 10. Assume that. Let. Then for-a.s., there exist random radiumand constantsuch that for all, the solutionof problem (23) withsatisfies that for every, whereis the collection of tempered random subsets of.
Proof. Using the classic Gronwall’s lemma (see ) to the inequality (42) on the intervalwith , we get that Note that by Lemma 9 there existssuch that for all, Then by (57) and (58) it gives that for alland, which gives an expression for.
Lemma 11. Assume that. Let. Then for-a.s.and every, there are, andsuch that for alland, the solution of problem (23) withsatisfies that whereis the collection of tempered random subsets of.
Proof . Multiplying (23) bywith respect to theinner product leads us to where In order to estimate, we rewrite it as where by utilizing the inequality (14) and Agmon’s inequality in(see ), it gives that Similarly by utilizing the Agmon’s inequality, we deduce that Then by (63)–(66) we get that We then estimatein (61). We first have where Then it follows from (68) and (69) that Moreover, Then by (67), (70), and (71), formula (61) becomes where Then by utilizing Lemma 7 to (72) we deduce that By Lemma 11, there existssuch that for all, where is independent of. By a similar calculation as (75), we find that there exist an random variablesuch that for all, whereis in Lemma 9. Then (74) together with (75) and (77) implies that for all, as. Consequently, for every, there exists a integerand positive constantssuch that for alland,