Existence of a martingale solution of the stochastic NavierStokes equations in unbounded 2D and 3Ddomains.
Abstract.
Stochastic NavierStokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical FaedoGalerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Moreover, some compactness and tightness criteria in nonmetric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.
Keywords: Stochastic NavierStokes equations, martingale solution, compactness method
AMS subject classification (2000): primary 35Q30, 60H15; secondary 76M35
1. Introduction.
Let be an open connected possibly unbounded subset with smooth boundary , where . We consider the stochastic NavierStokes equations
in , with the incompressibility condition
and with the homogeneous boundary condition . In this problem and represent the velocity and the pressure of the fluid. Furthermore, stands for the deterministic external forces and , where is a cylindrical Wiener process, stands for the random forces.
The above problem can be written in an abstract form as the following initial value problem
Here and are appropriate maps corresponding to the Laplacian and the nonlinear term, respectively in the NavierStokes equations, see Section 2. We impose rather general assumptions (G), (G) and (G) on the noise formulated in (A.3) in Section 4. These assumptions cover the following special case
where are independent real standard Brownian motions, see Section 6.
We prove the existence of a martingale solution. The construction of a solution is based on the classical FaedoGalerkin approximation, i.e.
given in Section 5. The crucial point is to prove suitable uniform a priori estimates on . Analogously to [21], we prove that the following estimates hold
and
for , where is given parameter, see Section 4. Here, and denote the closures in the Sobolev space and , respectively of the divergencefree vector fields of class with compact supports contained in . The solutions to the Galerkin scheme generate a sequence of laws on appropriate functional spaces. To prove that this sequence of probability measures is weakly compact we need appropriate tightness criteria.
In Section 3 we prove certain deterministic compactness results, see Lemmas 3.1 and 3.3. If is unbounded, then the embedding is not compact. However, using Lemma 2.5 from [20] (see Lemma C.1 in Appendix C), we can find a separable Hilbert space such that
the embedding being dense and compact. Then we have
where and are the dual spaces of and , respectively, being identified with and is the dual operator to the embedding . Moreover, is compact as well. Modifying the proof of the classical Dubinsky Theorem, [38, Theorem IV.4.1], we obtain a certain deterministic compactness criterion, see Lemma 3.1. Namely, we will show that a set is relatively compact in the intersection
if the following two conditions hold

, i.e. is bounded in ,

.
Here denotes the space of valued continuous functions,
is the space equipped with the weak topology and is a Fréchet space defined in (3.1) in Section 3. Let us notice that the second condition implies the equicontinuity of the family of valued functions. Thus the above two conditions are the same as in the Dubinsky Theorem. However, since the embedding is not compact, then in comparison to the Dubinsky Theorem we have the space instead of .
Next, using this version of the Dubinsky Theorem, we will prove another deterministic compactness criterion, see Lemma 3.3. Namely, we will show that a set is relatively compact in the intersection
if the following three conditions hold

,

,

.
These results were inspired by Lemma 2.7 due to Mikulevicius and Rozovskii [31], where the case of , is considered. In [31] the space is compactly embedded in the Fréchet space for sufficiently large . Then, the authors prove the deterministic compactness criterion in the intersection
The main difference with the approach of Mikulevicius and Rozovskii, is that instead of the Fréchet space , we consider the space dual to the Hilbert space constructed in a special way by Holly and Wiciak, see [20, Lemma 2.5]. This allows us to prove the above mentioned modification of the Dubinsky Theorem. The space will be also of crucial importance in further construction of a martingale solution.
Using Lemma 3.3 and the Aldous condition in the form given by Métivier [30], we obtain a new tightness criterion for the laws on the space , see Corollary 3.9. Next, we prove that the set of laws is tight on . The next step in our construction of a martingale solution differs from the approach of Mikulevicius and Rozovskii. We apply the method used by Da Prato and Zabczyk in [16, Chapter 8]. This method is based on the Skorokhod Theorem and the martingale representation Theorem. However, we will apply the Jakubowski’s version of the Skorokhod Theorem for nonmetric spaces in the form given by Brzeźniak and Ondreját [11], [23]. In [16, Chapter 8] the authors impose the linear growth conditions on the nonlinear term and assume the compactness of the appropriate semigroup. The assumptions considered in [16, Chapter 8] do not cover the stochastic NavierStokes equations, however, we can use the ideas introduced there. This method seems to us more direct. In the case of 2D domains we prove moreover the existence and uniqueness of strong solutions.
Stochastic Partial Differential Equations (SPDEs) can be viewed as an intersection of the infinite dimensional Stochastic Analysis and Partial Differential Equations. The theory of SPDEs began in the early ’s with works of BensoussanTemam [4], Dawson and Salehi [18] and many others. Due to contributions of several authors such as Pardoux [33] KrylovRozovskii [25], Da PratoZabczyk [16] many aspects of this new theory are now well developed and understood. The study of stochastic NSEs initiated in [4] was continued by many, for instance Brzeźniak et all [7, 8], Flandoli and Ga̧tarek [21], HairerMattingly [19] and MikuleviciusRozovskii [31]. In the last paper the authors study the existence of a martingale solution of the stochastic NavierStokes equations for turbulent flows in , () corresponding to the Kraichnan model of turbulence.
Stochastic NavierStokes equations in unbounded 2D and 3D domains with the noise independent on were considered by Capiński and Peszat [15] and Brzeźniak and Peszat [12]. The solutions are constructed in weighted spaces. Invariant measures for stochastic NavierStokes equations with an additive noise in some unbounded 2D domains are investigated by Brzeźniak and Li [9]. Our results generalize the corresponding existence results from [15], [9] [21], and [31]. What concerns modelling of noise, we have tried to be as general as possible and include the roughest noise possible. One should bear in mind that the rougher the noise the closer the model is to reality. Moreover, Landau and Lifshitz their fundamental 1959 work [26, Chapter 17] proposed to study NSEs under additional stochastic small fluctuations. Consequently these authors considered the classical balance laws for mass, energy and momentum forced by a random noise, to describe the fluctuations, in particular local stresses and temperature, which are not related to the gradient of the corresponding quantities. In [27, Chapter 12] the same authors found correlations for the random forcing by following the general theory of fluctuations. One of the requirements imposed on the noise is that it is either spatially uncorrelated or correlated as little as possible.
The present paper is organized as follows. In Section 2 we recall basic definitions and introduce some auxilliary operators. In Section 3 we are concerned with the compactness results. In Section 4, we formulate the NavierStokes problem as an abstract stochastic evolution equation. The main theorem about existence and construction of the martingale solution is in Section 5. Some auxilliary results connected with the proof are given in Appendices A and B. In Section 6, we consider some example of the noise. For the convenience of the reader, in Appendix C we recall Lemma 2.5 from [20] with the proof. Section 7 and Appendix D are devoted to 2D NavierStokes equations.
Acknowledgments
The authours would like to thank an anonymous referee for most useful queries and comments which led to the improvement of the paper.
2. Functional setting
2.1. Notations.
Let be two real normed spaces. The symbol stands for the space of all bounded linear operators from to . If , then is called the dual space of . The symbol denotes the standard duality pairing. If no confusion seems likely we omit the subscripts and write . If both spaces and are separable Hilbert, then by we will denote the Hilbert space of all HilbertSchmidt operators from to endowed with the standard norm.
Assume that are Banach spaces. Let be a densely defined linear operator from to and let denote the domain of . Let
Note that if is bounded. Let . Since is densely defined, the functional can be uniquely extended to the linear bounded functional . The operator defined by
is called the dual operator of .
Assume that are Hilbert spaces with scalar products and , respectively. Let be a densely defined linear operator. By we denote the adjoint operator of . In particular, , and
Note that if is bounded.
2.2. Basic definitions
Let be an open subset with smooth boundary , . Let and let denote the Banach space of Lebesgue measurable valued th power integrable functions on the set . The norm in is given by
By we denote the Banach space of Lebesgue measurable essentially bounded valued functions defined on . The norm is given by
If , then is a Hilbert space with the scalar product given by
Let stand for the Sobolev space of all for which there exist weak derivatives , . It is a Hilbert space with the scalar product given by
where
(2.1) 
Let denote the space of all valued functions of class with compact supports contained in and let
In the space we consider the scalar product and the norm inherited from and denote them by and , respectively, i.e.
In the space we consider the scalar product inherited from , i.e.
(2.2) 
where is defined in (2.1), and the norm the norm induced by , i.e.
(2.3) 
where .
Let us consider the following trilinear form
(2.4) 
We will recall the fundamental properties of the form . Since usually one considers the bounded domain case we want to recall only those results that are valid in unbounded domains as well.
By the Sobolev embedding Theorem and the Hőlder inequality, we obtain the following estimates
(2.5)  
(2.6) 
for some positive constant . Thus the form is continuous on , see also [35]. Moreover, if we define a bilinear map by , then by inequality (2.6) we infer that for all and that the following inequality holds
(2.7) 
Moreover, the mapping is bilinear and continuous.
Let us also recall the following properties of the form , see Temam [35], Lemma II.1.3,
(2.8) 
In particular,
(2.9) 
Let us, for any define the following standard scale of Hilbert spaces
If then by the Sobolev embedding Theorem,
Here denotes the space of continuous and bounded valued functions defined on . If and with then
for some constant . Thus, can be uniquely extented to the trilinear form (denoted by the same letter)
and for and . At the same time the operator can be uniquely extended to a bounded bilinear operator
In particular, it satisfies the following estimate
(2.10) 
We will also use the following notation, .
Lemma 2.1.
The map is locally Lipschitz continuous, i.e. for every there exists a constant such that
(2.11) 
Proof.
This is classical but for completness we provide the proof. The assertion follows from the following estimates
Thus the Lipschitz condition holds with , where stands for the norm of the bilinear map . The proof is thus complete. ∎
2.3. Some operators
Consider the natural embedding and its adjoint . Since the range of is dense in , the map is onetoone. Let us put
(2.12) 
Notice that for all and
(2.13) 
Indeed, this follows immediatelly from the following equalities
Let
where is defined by (2.1). Let us notice that if , then and
(2.14) 
Indeed, from (2.3) and the following inequalities
it follows that and inequality (2.14) holds. Denoting by the dual pairing between and , i.e. , we have the following equality
(2.15) 
Lemma 2.2.

For any and :
where stands for the identity operator on . In particular,
(2.16) 
is dense in .
Proof.
To prove assertion (a), let and . By (2.13), (2.2) and (2.15), we have
Let us move to the proof of part (b). Since is dense in , it is sufficient to prove that is dense in . Let be an arbitrary element orthogonal to with respect to the scalar product in . Then
On the other hand, by (a) and (2.2), for . Hence for . Since is onto, we infer that , which completes the proof. ∎
Let us assume that . It is clear that is dense in and the embedding is continuous. Then by Lemma C.1 in Appendix C, there exists a Hilbert space such that , is dense in and
(2.17) 
Then we have
(2.18) 
Since the embedding is compact, is compact as well. Consider the composition
and its adjoint
Note that is compact and since the range of is dense in , is onetoone. Let us put
(2.19) 
It is clear that is onto. Let us also notice that
(2.20) 
Indeed, by (2.19) we have for all and
which proves (2.20). By equality (2.20) and the densiness of in , we infer similarly as in the proof of assertion (b) in Lemma 2.2 that is dense in .
Let us also put
(2.21) 
and
(2.22) 
The operators and are densely defined and onto. Let us also notice that
(2.23) 
Since is selfadjoint and is compact, there exists an orthonormal basis of composed of the eigenvectors of operator . Let be the eigenvalue corresponding to , i.e.
(2.24) 
Notice that , , because . Let us fix and let denote the linear space spanned by the vectors . Let be the operator from to defined by
(2.25) 
We will consider the restriction of the operator to the space denoted still by . More precisely, we have , i.e. every element induces a functional by the formula
Thus the restriction of to is given by
(2.26) 
Hence in particular, is the orthogonal projection from onto . Restrictions of to other spaces considered in (2.18) will also be denoted by .
Lemma 2.3.
The following equality holds
(2.27) 
Proof.
Let us denote
Lemma 2.4.

The system is the orthonormal basis in the space . Moreover,
(2.28) 
For every and
(2.29) i.e. the restriction of to is the orthogonal projection onto .

For every and we have

,

,

.

Proof.
(2.30) 
where if and if . In particular, equality (2.28) holds. By (2.30) and (2.28) the system is orthonormal in . To prove that it is a basis in let be an arbitrary vector orthogonal to each , . By (2.20) and (2.24) we obtain the following equalities
Since ,
(2.31) 
Since is the orthonormal basis in , we infer that . This means that the system is the orthonormal basis in .
To prove assertion (b) let us fix . By (2.28), (2.24) and (2.20), we have
(2.32) 
Since is the orthonormal basis in , the restriction of to the space is the orthogonal projection onto .
Assertion (c) follows immediately from (b) and the continuity of the embeddings
This completes the proof of the Lemma. ∎
3. Compactness results.
3.1. Deterministic compactness criteria
Let us choose . We have the following sequence of Hilbert spaces
(3.1) 
with the embedding being compact. In particular,
(3.2) 
and is compact as well. Let be a sequence of bounded open subsets of with regular boundaries such that and . We will consider the following spaces of restrictions of functions defined on to subsets , i.e.
(3.3) 
with appropriate scalar products and norms, i.e.
and for and for . The symbols and will stand for the corresponding dual spaces.
Since the sets are bounded,
the embeddings are compact.  (3.4) 