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Abstract and Applied Analysis
VolumeΒ 2009, Article IDΒ 723236, 27 pages
http://dx.doi.org/10.1155/2009/723236
Research Article

On the Stochastic 3D Navier-Stokes-𝛼 Model of Fluids Turbulence

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa

Received 24 June 2009; Accepted 21 October 2009

Academic Editor: RuedigerΒ Landes

Copyright Β© 2009 Gabriel Deugoue and Mamadou Sango. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the stochastic 3D Navier-Stokes-𝛼 model which arises in the modelling of turbulent flows of fluids. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. The adequate notion of solutions is that of probabilistic weak solution. We establish the existence of a such of solution. We also discuss the uniqueness.

1. Introduction

In this paper, we are interested in the study of probabilistic weak solutions of the 3D Navier-Stokes-𝛼(NSβˆ’π›Ό) model (also known as the Lagrangien averaged Navier-Stokes-alpha model or the viscous Camassa-Holm equations) with homogeneous Dirichlet boundary conditions in a bounded domains in the case in which random perturbations appear. To be more precise, let 𝐷 be a connected and bounded open subset of 𝑅3 with 𝐢2 boundary πœ•π· and a final time 𝑇>0. We denote by 𝐴 the Stokes operator and consider the system

πœ•π‘‘(π‘’βˆ’π›ΌΞ”π‘’)+𝜈(π΄π‘’βˆ’π›ΌΞ”(𝐴𝑒))+(π‘’β‹…βˆ‡)(π‘’βˆ’π›ΌΞ”π‘’)βˆ’π›Όβˆ‡π‘’βˆ—β‹…Ξ”π‘’+βˆ‡π‘=𝐹(𝑑,𝑒)+𝐺(𝑑,𝑒)π‘‘π‘Š,𝑑𝑑in𝐷×(0,𝑇),βˆ‡β‹…π‘’=0,in𝐷×(0,𝑇),𝑒=0,𝐴𝑒=0,onπœ•DΓ—(0,T),𝑒(0)=𝑒0,in𝐷,(1.1) where 𝑒=(𝑒1,𝑒2,𝑒3) and 𝑝 are unknown random fields on 𝐷×(0,𝑇), representing, respectively, the large-scale velocity and the pressure, in each point of 𝐷×(0,𝑇). The constant 𝜈>0 and 𝛼>0 are given, and represent, respectively, the kinematic viscosity of the fluid, and the square of the spatial scale at which fluid motion is filtered. The terms 𝐹(𝑑,𝑒) and 𝐺(𝑑,𝑒)(π‘‘π‘Š/𝑑𝑑) are external forces depending on 𝑒, where π‘Š is an π‘…π‘š-valued standard Wiener process. Finally, 𝑒0 is a given nonrandom velocity field.

The deterministic version of (1.1), that is, when 𝐺=0 has been the object of intense investigations over the last years [1–5] and the initial motivation was to find a closure model for the 3D turbulence-averaged Reynolds model. A key interest in the model is the fact that it serves as a good approximation to the 3D Navier-Stokes equations. One of the main reasons justifying its use is the high computational cost that the Navier-Stokes model requires. Many important results have been obtained in the deterministic case. More precisely, the global well posedness of weak solutions for the NS-𝛼 model on bounded domains has been established in [6, 7] amongst others, and the asymptotic behavior can be found in [6]. Similar results have been proved by Foias et al. [8] in the case of periodic boundary conditions.

However, in order to consider a more realistic model our problem, it is sensible to introduce some kind of noise in the equations. This may reflect, some environmental effects on the phenomena, some external random forces, and so forth. To the best of our knowledge, the existence and uniqueness of solutions of the stochastic version (1.1) which we consider in this paper has only been analyzed in [9] (see also [10]) in the case of Lipschitz assumptions on 𝐹 and 𝐺. The case of non-Lipschitz assumptions on the coefficients 𝐹 and 𝐺 is the main concern of the present paper. This question has been opened till now. The general motivation for studying weak rather than strong solutions of stochastic equations is that existence of weak solutions can be carried through under weaker regularity on the coefficients. This was pointed out, for instance, in [11].

In this paper, we will establish the existence of probabilistic weak solutions for the problem (1.1) under appropriate conditions on the data. Under the strong assumptions on 𝐹 and 𝐺, we prove the uniqueness of weak solutions. The method used for the proof of our existence results is different from the method in [9]. To prove the existence, we use the Galerkin approximation method employing special bases, combined with some famous theorems of probabilistic nature due to Prokhorov [12] and Skorokhod [13].

The paper is organized as follows. In Section 2, we establish some properties of nonlinear term appearing in our equations. The rigorous statement of our problem as well as the main results are included in Section 3 and we show how our problem can be reformulated as an abstract stochastic model. Section 4 is devoted to the proof of our main results.

2. Properties of the Nonlinear Terms in (1.1)

Following [9], we establish some properties of the nonlinear term (π‘’β‹…βˆ‡)(π‘’βˆ’π›ΌΞ”π‘’)βˆ’π›Όβˆ‡π‘’βˆ—β‹…Ξ”π‘’ appearing in (1.1).

We denote by (β‹…,β‹…) and |β‹…|, respectively, the scalar product and associated norm in (𝐿2(𝐷))3, and by (βˆ‡π‘’,βˆ‡π‘£), the scalar product in ((𝐿2(𝐷))3)3 of the gradients of 𝑒 and 𝑣. We consider the scalar product in (𝐻10(𝐷))3 defined by

𝐻((𝑒,𝑣))=(𝑒,𝑣)+𝛼(βˆ‡π‘’,βˆ‡π‘£),𝑒,π‘£βˆˆ10ξ€Έ(𝐷)3,(2.1) where its associated norm β€–β‹…β€– is, in fact, equivalent to the usual gradient norm. We denote by 𝐻 the closure in (𝐿2(𝐷))3 of the set

𝒱=π‘£βˆˆ(π’Ÿ(𝐷))3βˆΆβˆ‡β‹…π‘£=0in𝐷,(2.2) and by 𝑉 the closure of 𝒱 in (𝐻10(𝐷))3. Then 𝐻 is a Hilbert space equipped with the inner product of (𝐿2(𝐷))3, and 𝑉 is a Hilbert subspace of (𝐻10(𝐷))3.

Denote by 𝐴 the Stokes operator, with domain 𝐷(𝐴)=(𝐻2(𝐷))3βˆ©π‘‰, defined by

𝐴𝑀=βˆ’π’«(Δ𝑀),π‘€βˆˆπ·(𝐴),(2.3) where 𝒫 is the projection operator from (𝐿2(𝐷))3 onto 𝐻. Recall that as πœ•π· is 𝐢2,|𝐴𝑀| defines in 𝐷(𝐴) a norm which is equivalent to the (𝐻2(𝐷))3 norm, that is, there exists a constant 𝑐1(𝐷), depending only on 𝐷, such that

‖𝑀‖(𝐻2(𝐷))3≀𝑐1||||(𝐷)𝐴𝑀,βˆ€π‘€βˆˆπ·(𝐴),(2.4) and so 𝐷(𝐴) is a Hilbert space with respect to the scalar product

(𝑣,𝑀)𝐷(𝐴)=(𝐴𝑣,𝐴𝑀).(2.5) For π‘’βˆˆπ·(𝐴) and π‘£βˆˆ(𝐿2(𝐷))3, we define (π‘’β‹…βˆ‡)𝑣 as the element of (π»βˆ’1(𝐷))3 given by

⟨(π‘’β‹…βˆ‡)𝑣,π‘€βŸ©βˆ’1=3𝑖,𝑗=1ξ«πœ•π‘–π‘£π‘—,π‘’π‘–π‘€π‘—ξ¬βˆ’1𝐻,βˆ€π‘€βˆˆ10ξ€Έ(𝐷)3,(2.6) where by βŸ¨π‘’,π‘£βŸ©βˆ’1, we denote either the duality product between (π»βˆ’1(𝐷))3 and (𝐻10(𝐷))3 or between π»βˆ’1(𝐷) and 𝐻10(𝐷).

Observe that (2.6) is meaningful, since 𝐻2(𝐷)βŠ‚πΏβˆž(𝐷) and 𝐻10(𝐷)βŠ‚πΏ6(𝐷) with continuous injections. This implies that π‘’π‘–π‘€π‘—βˆˆπ»10(𝐷), and there exists a constant 𝑐2(𝐷)>0, depending only on 𝐷, such that

||⟨(π‘’β‹…βˆ‡)𝑣,π‘€βŸ©βˆ’1||≀𝑐2||||𝐿(𝐷)𝐴𝑒|𝑣|‖𝑀‖,βˆ€(𝑒,𝑣,𝑀)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3×𝐻10ξ€Έ(𝐷)3.(2.7) Observe also that if π‘£βˆˆ(𝐻1(𝐷))3, then the definition above coincides with the definition of (π‘’β‹…βˆ‡)𝑣 as the vector function whose components are βˆ‘3𝑖=1π‘’π‘–πœ•π‘–π‘£π‘—, for 𝑗=1,2,3. However, as it not known whether the solutions of the stochastic problem (1.1) have the same regularity as the deterministic case (we only can ensure 𝐻2 instead of 𝐻3), the present extension is necessary.

Now, if π‘’βˆˆπ·(𝐴), then βˆ‡π‘’βˆ—βˆˆ(𝐻1(𝐷))3Γ—3βŠ‚(𝐿6(𝐷))3Γ—3, and consequently, for π‘£βˆˆ(𝐿2(𝐷))3, we have that βˆ‡π‘’βˆ—β‹…π‘£βˆˆ(𝐿3/2(𝐷))3βŠ‚(π»βˆ’1(𝐷))3, with

βŸ¨βˆ‡π‘’βˆ—β‹…π‘£,π‘€βŸ©βˆ’1=3𝑖,𝑗=1ξ€œπ·ξ€·πœ•π‘—π‘’π‘–ξ€Έπ‘£π‘–π‘€π‘—ξ€·π»π‘‘π‘₯,βˆ€π‘€βˆˆ10ξ€Έ(𝐷)3.(2.8) It follows that there exists a constant 𝑐3(𝐷), depending only on 𝐷, such that

||βŸ¨βˆ‡π‘’βˆ—β‹…π‘£,π‘€βŸ©βˆ’1||≀𝑐3||||𝐿(𝐷)𝐴𝑒|𝑣|‖𝑀‖,βˆ€(𝑒,𝑣,𝑀)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3×𝐻10ξ€Έ(𝐷)3.(2.9) We have the following results.

Proposition 2.1. For all (𝑒,𝑀)∈𝐷(𝐴)×𝐷(𝐴) and for all π‘£βˆˆ(𝐿2(𝐷))3, it follows that ⟨(π‘’β‹…βˆ‡)𝑣,π‘€βŸ©βˆ’1=βˆ’βŸ¨βˆ‡π‘€βˆ—β‹…π‘£,π‘’βŸ©βˆ’1.(2.10)

Proof. If (𝑒,𝑀)∈𝐷(𝐴)×𝐷(𝐴), then for each 𝑖,𝑗=1,2,3, we have π‘’π‘–π‘€π‘—βˆˆπ»10(𝐷) and consequently ξ«πœ•π‘–π‘£π‘—,π‘’π‘–π‘€π‘—ξ¬βˆ’1ξ€œ=βˆ’π·π‘£π‘—πœ•π‘–ξ€·π‘’π‘–π‘€π‘—ξ€Έξ€œπ‘‘π‘₯=βˆ’π·π‘£π‘—π‘€π‘—πœ•π‘–π‘’π‘–ξ€œπ‘‘π‘₯βˆ’π·π‘£π‘—π‘’π‘–πœ•π‘–π‘€π‘—π‘‘π‘₯,(2.11) using βˆ‡β‹…π‘’=0, we have (2.10).

Consider now the bilinear form defined by

π‘βˆ—(𝑒,𝑣,𝑀)=⟨(π‘’β‹…βˆ‡)𝑣,π‘€βŸ©βˆ’1+βŸ¨βˆ‡π‘’βˆ—β‹…π‘£,π‘€βŸ©βˆ’1,𝐿(𝑒,𝑣,𝑀)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3×𝐻10ξ€Έ(𝐷)3.(2.12)

Proposition 2.2. The bilinear form π‘βˆ— satisfies π‘βˆ—(𝑒,𝑣,𝑀)=βˆ’π‘βˆ—ξ€·πΏ(𝑀,𝑣,𝑒),βˆ€(𝑒,𝑣,𝑀)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3×𝐷(𝐴),(2.13) and consequently, π‘βˆ—ξ€·πΏ(𝑒,𝑣,𝑒)=0,βˆ€(𝑒,𝑣)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3.(2.14) Moreover, there exists a constant 𝑐(𝐷)>0, depending only on 𝐷, such that ||π‘βˆ—||||||𝐿(𝑒,𝑣,𝑀)≀𝑐(𝐷)𝐴𝑒|𝑣|‖𝑀‖,βˆ€(𝑒,𝑣,𝑀)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3×𝐻10ξ€Έ(𝐷)3,||π‘βˆ—||||||𝐿(𝑒,𝑣,𝑀)≀𝑐(𝐷)‖𝑒‖|𝑣|𝐴𝑀,βˆ€(𝑒,𝑣,𝑀)∈𝐷(𝐴)Γ—2ξ€Έ(𝐷)3×𝐷(𝐴).(2.15) Thus, in particular, π‘βˆ— is continuous on 𝐷(𝐴)Γ—(𝐿2(𝐷))3Γ—(𝐻10(𝐷))3.

Proof. The proof is straightforward consequences of (2.7), (2.9), and (2.10).

3. Statement of the Problem and the Main Results

We now introduce some probabilistic evolutions spaces.

Let (Ξ©,𝐹,{𝐹𝑑}0≀𝑑≀𝑇,𝑃) be a filtered probability space and let 𝑋 be a Banach space. For π‘Ÿ,π‘žβ‰₯1, we denote by

𝐿𝑝(Ξ©,𝐹,𝑃;πΏπ‘Ÿ(0,𝑇;𝑋))(3.1) the space of functions 𝑒=𝑒(π‘₯,𝑑,πœ”) with values in 𝑋 defined on [0,𝑇]Γ—Ξ© and such that

(1)𝑒 is measurable with respect to (𝑑,πœ”) and for almost all 𝑑, 𝑒 is 𝐹𝑑 measurable, (2)‖𝑒‖𝐿𝑝(Ξ©,𝐹,𝑃;πΏπ‘Ÿ(0,𝑇;𝑋))=ξƒ¬πΈξ‚΅ξ€œπ‘‡0β€–π‘’β€–π‘Ÿπ‘‹ξ‚Άπ‘‘π‘‘π‘/π‘Ÿξƒ­1/π‘Ÿ<∞,(3.2) where 𝐸 denotes the mathematical expectation with respect to the probability measure 𝑃.

The space 𝐿𝑝(Ξ©,𝐹,𝑃;πΏπ‘Ÿ(0,𝑇;𝑋)) so defined is a Banach space.

When π‘Ÿ=∞, the norm in 𝐿𝑝(Ξ©,𝐹,𝑃;𝐿∞(0,𝑇;𝑋)) is given by

‖𝑒‖𝐿𝑝(Ξ©,𝐹,𝑃;𝐿∞(0,𝑇;𝑋))=𝐸supess0≀𝑑≀𝑇‖𝑒‖𝑝𝑋1/𝑝.(3.3)

We make precise our assumptions on (1.1).

We start with the nonlinear function 𝐹 and 𝐺. We assume that

ξ€·π»πΉβˆΆ(0,𝑇)Γ—π‘‰βŸΆβˆ’1ξ€Έ(𝐷)3,ameasurable.e.𝑑,𝑒↣𝐹(𝑑,𝑒)∢continuousfrom𝑉toξ€·π»βˆ’1ξ€Έ(𝐷)3‖𝐹(𝑑,𝑒)β€–ξ€·π»βˆ’1ξ€Έ(𝐷)3≀𝑐(1+‖𝑒‖),𝐺∢(0,𝑇)Γ—π‘‰βŸΆ(𝐿2(𝐷))3ξ€Έπ‘š,ameasurable.e.𝑑,𝑒↣𝐺(𝑑,𝑒)∢continuousfrom𝑉to𝐿2ξ€Έ(𝐷)3ξ‚π‘š||||𝐺(𝑑,𝑒)((𝐿2(𝐷))3)π‘šβ‰€π‘(1+‖𝑒‖).(3.4) We will define the concept of weak solution of the problem (1.1), namely, the following.

Definition 3.1. A weak solution of (1.1) means a system (Ξ©,β„±,{ℱ𝑑}0≀𝑑≀𝑇,𝒫,π‘Š,𝑒) such that (1)(Ξ©,β„±,𝒫) is a probability space, ({ℱ𝑑},0≀𝑑≀𝑇) is a filtration, (2)π‘Š is an π‘š-dimensional {ℱ𝑑}  standard Wiener process, (3)𝑒(𝑑) is ℱ𝑑 adapted for all π‘‘βˆˆ[0,𝑇]βˆΆπ‘’βˆˆπΏπ‘ξ€·Ξ©,β„±,𝒫;𝐿2ξ€Έ(0,𝑇,𝐷(𝐴))βˆ©πΏπ‘(Ξ©,β„±,𝒫;𝐿∞(0,𝑇,𝑉)),βˆ€1≀𝑝<∞,(3.5)(4)for almost all π‘‘βˆˆ(0,𝑇), the following equation holds 𝒫-a.s.ξ€œ((𝑒(𝑑),Ξ¦))+πœˆπ‘‘0(ξ€œπ‘’(𝑠)+𝛼𝐴𝑒(𝑠),𝐴Φ)𝑑𝑠+𝑑0π‘βˆ—(=𝑒𝑒(𝑠),𝑒(𝑠)βˆ’π›ΌΞ”π‘’(𝑠),Ξ¦)𝑑𝑠0+ξ€œ,Φ𝑑0⟨𝐹(𝑠,𝑒(𝑠)),Ξ¦βŸ©βˆ’1ξ‚΅ξ€œπ‘‘π‘ +𝑑0𝐺(𝑠,𝑒(𝑠))π‘‘π‘Š(𝑠),Ξ¦(3.6) for all Φ∈𝐷(𝐴).

Our two major results are as follows.

Theorem 3.2 (Existence). Assume (3.4) and 𝑒0βˆˆπ‘‰. Then there exists a weak solution (Ξ©,β„±,{ℱ𝑑}0≀𝑑≀𝑇,𝒫,π‘Š,𝑒) of (1.1) in the sense of Definition 3.1.
Moreover π‘’βˆˆπΏπ‘(Ξ©,β„±,𝒫;𝐢([0,𝑇];𝑉)), and there exists a unique Μƒπ‘βˆˆπΏ2(Ξ©,ℱ𝑑,𝒫;π»βˆ’1(0,𝑑;π»βˆ’1(𝐷)), for all π‘‘βˆˆ[0,𝑇], such that 𝒫-a.s.
πœ•π‘‘(π‘’βˆ’π›ΌΞ”π‘’)+𝜈(π΄π‘’βˆ’π›ΌΞ”(𝐴𝑒))+(π‘’β‹…βˆ‡)(π‘’βˆ’π›ΌΞ”π‘’)βˆ’π›Όβˆ‡π‘’βˆ—β‹…Ξ”π‘’+βˆ‡Μƒπ‘=𝐹(𝑑,𝑒)+𝐺(𝑑,𝑒)π‘‘π‘Š,𝑑𝑑inξ€·π’Ÿξ…žξ€Έ((0,𝑇)×𝐷)3,ξ€œπ·Μƒπ‘π‘‘π‘₯=0,inπ’Ÿξ…ž(0,𝑇),(3.7) where 𝐺(𝑑,𝑒)(π‘‘π‘Š/𝑑𝑑) denotes the time derivative of βˆ«π‘‘0𝐺(𝑠,𝑒(𝑠))π‘‘π‘Šπ‘ , that is, by definition 𝐺(𝑑,𝑒)π‘‘π‘Šπ‘‘π‘‘=πœ•π‘‘ξ‚΅ξ€œ.0𝐺(𝑠,𝑒(𝑠))π‘‘π‘Šπ‘ ξ‚Ά,inπ’Ÿξ…žξ‚€ξ€·πΏ0,𝑇;2ξ€Έ(𝐷)3,𝒫-a.s.(3.8)

Corollary 3.3 (Uniqueness). Assume that 𝐹 and 𝐺 are Lipschitz with respect to the second variable 𝑒0βˆˆπ‘‰. Then there exists a unique weak solution of problem (1.1) in the sense of Definition 3.1.
Moreover, two strong solutions on the same Brownian stochastic basis coincide a.s.

3.1. Formulation of Problem (1.1) as an Abstract Problem

We will rewrite our model as an abstract problem.

We identify 𝑉 with its topological dual π‘‰ξ…ž and we have the Gelfand triple 𝐷(𝐴)βŠ‚π‘‰βŠ‚π·(𝐴)ξ…ž.

We denote by βŸ¨β‹…,β‹…βŸ© the duality product between 𝐷(𝐴)ξ…ž and 𝐷(𝐴). We define

βŸ¨ξ‚π΄π‘’,π‘£βŸ©=𝜈(𝐴𝑒,𝑣)+πœˆπ›Ό(𝐴𝑒,𝐴𝑣),𝑒,π‘£βˆˆπ·(𝐴).(3.9) It is clear that for all π‘£βˆˆπ·(𝐴),

||||2βŸ¨π΄π‘’,π‘£βŸ©=2𝜈(𝐴𝑣,𝑣)+2πœˆπ›Ό(𝐴𝑣,𝐴𝑣)β‰₯2πœˆπ›Όπ΄π‘£2,(3.10) and, if we denote by πœ†π‘˜ and π‘€π‘˜,π‘˜β‰₯1, the eigenvalues, and their corresponding eigenvalues associated to 𝐴, then

βŸ¨ξ‚π΄π‘€π‘˜,π‘£βŸ©=πœˆπœ†π‘˜π‘€ξ€·ξ€·π‘˜.,𝑣(3.11) Thus, taking

𝛼=2πœˆπ›Ό,(3.12) we have

(a)𝐴 is a linear continuous operator ξ‚π΄βˆˆβ„’(𝐷(𝐴),𝐷(𝐴)ξ…ž), such that (a𝐴1)isself-adjoint(a2)thereexists𝛼>0,suchthat2𝐴𝑣,𝑣β‰₯𝛼‖𝑣‖2𝐷(𝐴),βˆ€π‘£βˆˆπ·(𝐴).(3.13) On the other hand, denote βŸ¨ξ‚π΅(𝑒,𝑣),π‘€βŸ©=π‘βˆ—ξ‚(𝑒,π‘£βˆ’π›ΌΞ”π‘£,𝑀),(𝑒,𝑣,𝑀)∈𝐷(𝐴)×𝐷(𝐴)×𝐷(𝐴),𝐹(𝑑,𝑒),𝑀=⟨𝐹(𝑑,𝑒),π‘€βŸ©βˆ’1,(𝑒,𝑀)βˆˆπ‘‰Γ—π‘‰.(3.14) Thus it is straightforward to check that if we take 𝑐1=(1+𝛼)𝑐1(𝐷)𝑐(𝐷),(3.15) then we obtain that (b)ξ‚π΅βˆΆπ·(𝐴)×𝐷(𝐴)→𝐷(𝐴)ξ…ž is a bilinear mapping such that (b1𝐡)⟨(𝑒,𝑣),π‘’βŸ©=0,βˆ€π‘’,π‘£βˆˆπ·(𝐴),(3.16)(b‖‖‖‖2)𝐡(𝑒,𝑣)𝐷(𝐴)′≀𝑐1‖𝑒‖‖𝑣‖𝐷(𝐴),βˆ€π‘’,π‘£βˆˆπ·(𝐴)×𝐷(𝐴),(3.17)(b||βŸ¨ξ‚||3)𝐡(𝑒,𝑣),π‘€βŸ©β‰€π‘1‖𝑒‖𝐷(𝐴)‖𝑣‖𝐷(𝐴)‖𝑀‖,βˆ€π‘’,𝑣,π‘€βˆˆπ·(𝐴).(3.18)(c)ξ‚πΉβˆΆ(0,𝑇)×𝑉→𝑉, measurable such that (c1)a.e𝐹.𝑑,𝑒↣(𝑑,𝑒)∢continuousfrom𝑉to𝑉(c‖‖‖‖2)𝐹(𝑑,𝑒)≀𝑐(1+‖𝑒‖).(3.19) Now, let 𝐼 denote the identity operator in 𝐻, and define 𝐺(𝑑,𝑒) as 𝐺(𝑑,𝑒)=(𝐼+𝛼𝐴)βˆ’1βˆ˜π’«βˆ˜πΊ(𝑑,𝑒),π‘’βˆˆπ‘‰.(3.20)𝐼+𝛼𝐴 is bijective from 𝐷(𝐴) onto 𝐻, and ξ€·ξ€·(𝐼+𝛼𝐴)βˆ’1𝑓,𝑀=(𝑓,𝑀),βˆ€π‘“βˆˆπ»,π‘€βˆˆπ‘‰.(3.21) Thus, for eachπ‘“βˆˆπ»,β€–β€–(𝐼+𝛼𝐴)βˆ’1𝑓‖‖2||𝑓||=(𝑓,𝑒)≀|𝑒|,(3.22) where 𝑒=(𝐼+𝛼𝐴)βˆ’1𝑓, that is, (𝑒,π‘€π‘˜)+𝛼(𝐴𝑒,π‘€π‘˜)=(𝑓,π‘€π‘˜), for all π‘˜β‰₯1, so (1+π›Όπœ†π‘˜)(𝑒,π‘€π‘˜)=(𝑓,π‘€π‘˜), which implies 𝑒,π‘€π‘˜ξ€Έ=1ξ€·1+π›Όπœ†π‘˜ξ€Έξ€·π‘“,π‘€π‘˜ξ€Έβ‰€11+π›Όπœ†1𝑓,π‘€π‘˜ξ€Έ,|𝑒|2=βˆžξ“π‘˜=1𝑒,π‘€π‘˜ξ€Έ2≀1ξ€·1+π›Όπœ†1ξ€Έ2βˆžξ“π‘˜=1𝑓,π‘€π‘˜ξ€Έ2=1ξ€·1+π›Όπœ†1ξ€Έ2||𝑓||2.(3.23) Therefore, β€–β€–(𝐼+𝛼𝐴)βˆ’1𝑓‖‖2≀11+π›Όπœ†1||𝑓||2,(3.24) and, consequently, taking 𝑐̃𝑐=√1+π›Όπœ†1,(3.25) we obtain that (d)ξ‚πΊβˆΆ(0,𝑇)Γ—π‘‰β†’π‘‰βŠ—π‘š, measurable such that (d1)a.e.𝑑,𝑒↣𝐺(𝑑,𝑒)∢continuousfrom𝑉toπ‘‰βŠ—π‘š(d‖‖‖‖2)𝐺(𝑑,𝑒)π‘‰βŠ—π‘šβ‰€Μƒπ‘(1+‖𝑒‖),(3.26)

where π‘‰βŠ—π‘šπ‘–π‘ π‘‘β„Žπ‘’π‘π‘Ÿπ‘œπ‘‘π‘’π‘π‘‘π‘œπ‘“π‘šπ‘π‘œπ‘π‘–π‘’π‘ π‘œπ‘“π‘‰. Next, for each 𝑗β‰₯1, and all (𝑑,𝑒,Ξ¦)∈(0,𝑇)×𝑉×𝐷(𝐴), we have

=,(𝐺(𝑑,𝑒),Ξ¦)=(𝐼+𝛼𝐴)𝐺(𝑑,𝑒),Φ𝐺(𝑑,𝑒),Φ(3.27) and, for all π‘’βˆˆπΏ2(Ξ©,β„±,𝒫;𝐿∞(0,𝑇;𝑉)), (𝑑,Ξ¦)∈(0,𝑇)×𝐷(𝐴), it follows that

ξ‚΅ξ€œπ‘‘0ξ‚Ά=𝐺(𝑠,𝑒(𝑠))π‘‘π‘Š(𝑠),Φ𝑑𝑗=1ξ€œπ‘‘0𝐺𝑗(𝑠,𝑒(𝑠)),Ξ¦π‘‘π‘Šπ‘—=(𝑠)𝑑𝑗=1ξ€œπ‘‘0𝐺𝑗(𝑠,𝑒(𝑠),Ξ¦)π‘‘π‘Šπ‘—=ξ€œ(𝑠)𝑑0.𝐺(𝑠,𝑒(𝑠))π‘‘π‘Š(𝑠),Ξ¦ξ‚Άξ‚Ά(3.28) Consequently, in this abstract framework, a weak solution (Ξ©,β„±,{ℱ𝑑}0≀𝑑≀𝑇,𝒫,π‘Š,𝑒) of (1.1) is equivalently as follows.

Definition 3.4. It holds that(1)(Ξ©,β„±,𝒫) is a probability space, ({ℱ𝑑},0≀𝑑≀𝑇) is a filtration, (2)π‘Š is a π‘š-dimensional {ℱ𝑑} standard Wiener process, (3)𝑒(𝑑) is ℱ𝑑 adapted for all π‘‘βˆˆ[0,𝑇]π‘’βˆˆπΏπ‘ξ€·Ξ©,β„±,𝒫;𝐿2ξ€Έ(0,𝑇,𝐷(𝐴))βˆ©πΏπ‘(Ξ©,β„±,𝒫;𝐿∞(0,𝑇,𝑉)),βˆ€1≀𝑝<∞,(3.29)(4)for almost all π‘‘βˆˆ(0,𝑇), the following equation holds 𝒫-a.s.ξ€œπ‘’(𝑑)+𝑑0ξ‚ξ€œπ΄π‘’(𝑠)𝑑𝑠+𝑑0𝐡(𝑒(𝑠),𝑒(𝑠))𝑑𝑠=𝑒0+ξ€œπ‘‘0ξ‚ξ€œπΉ(𝑠,𝑒(𝑠))𝑑𝑠+𝑑0𝐺(𝑠,𝑒(𝑠))π‘‘π‘Š(𝑠),(3.30) as an equality in 𝐷(𝐴)ξ…ž.

Remark 3.5. However, (3.30) implies that π‘’βˆˆπ’ž(0,𝑇;𝐷(𝐴)ξ…ž), then 𝑒 is weakly continuous in 𝑉 [14, page 263] and the initial condition is meaningful.

4. Proofs of the Main Results

4.1. Proof of Theorem 3.2

We make use of the Galerkin approximation combined with the method of compactness.

We will split the proof into six steps.

4.1.1. Step  1. Construction of an Approximating Sequence

As the injection 𝐷(𝐴)β†ͺ𝑉 is compact, consider an orthonormal basis {𝑒𝑗}𝑗=1,2,…in𝐷(𝐴) which is orthogonal in 𝑉 such that 𝑒𝑗 are eigenfunctions of the spectral problem

𝑒𝑗,𝑣𝐷(𝐴)=πœ†π‘—π‘’ξ€·ξ€·π‘—,𝑣,βˆ€π‘£βˆˆπ·(𝐴),(4.1) where (.,.)𝐷(𝐴) denotes the scalar product in 𝐷(𝐴). For each π‘βˆˆβ„•, let 𝑉𝑁 be the span of {𝑒1,…,𝑒𝑁}.

Consider the probabilistic system

ξ‚΅Ξ©,𝐹,𝐹𝑑0≀𝑑≀𝑇,𝑃,π‘Šξ‚Ά.(4.2) We denote by 𝐸 the mathematical expectation with respect to (Ξ©,𝐹,𝑃).

We look for a sequence of functions 𝑒𝑁(𝑑) in 𝑉𝑁, that is,

𝑒𝑁(𝑑)=𝑁𝑗=1𝑐𝑁𝑗𝑑,πœ”ξ€Έπ‘’π‘—(π‘₯),(4.3) solutions of the following stochastic ordinary differential equations in π‘‰π‘βˆΆ

𝑑𝑒𝑁,𝑒𝑗+ξ‚€βŸ¨ξ‚ξ€Έξ€Έπ΄π‘’π‘(𝑑),π‘’π‘—ξ‚π΅ξ€·π‘’βŸ©+βŸ¨π‘(𝑑),𝑒𝑁(𝑑),π‘’π‘—βŸ©ξ‚=𝐹𝑑𝑑𝑑,𝑒𝑁(𝑑),𝑒𝑗𝐺𝑑𝑑+𝑑,𝑒𝑁(𝑑),π‘’π‘—π‘‘ξ‚ξ‚π‘’π‘Š,𝑗=1,2,…,𝑁𝑁(0)=𝑒𝑁0,(4.4) where 𝑒𝑁0βˆˆπ‘‰π‘ and is chosen with the requirements that

𝑒𝑁0βŸΆπ‘’0in𝑉asπ‘βŸΆβˆž.(4.5) There exists a a maximal solution to (4.4), that is, a stopping time 𝑇𝑁≀𝑇 such that (4.4) holds for 𝑑<𝑇𝑁 [11]. Solvability over (0,𝑇) will follow from a priori estimates for 𝑒𝑁 that we derive in the following section.

We have the following Fourier expansion:

𝑒𝑁(𝑑)=𝑁𝑗=1𝑒𝑁(𝑑),𝑒𝑗𝐷(𝐴)𝑒𝑗=𝑁𝑗=1πœ†π‘—π‘’ξ€·ξ€·π‘(𝑑),𝑒𝑗𝑒𝑗,‖‖𝑒𝑁‖‖(𝑑)2=𝑁𝑗=1πœ†π‘—π‘’ξ€·ξ€·π‘(𝑑),𝑒𝑗2.(4.6)

4.1.2. Step  2. A Priori Estimates

Throughout 𝐢 and 𝐢𝑖(𝑖=1,…) denotes a positive constant independent of 𝑁.

We have the following Lemma.

Lemma 4.1. It holds that 𝑒𝑁 satisfies the following a priori estimates: 𝐸sup0≀𝑑≀𝑇‖‖𝑒𝑁‖‖(𝑠)2+2ξ‚π›ΌπΈξ€œπ‘‡0‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)𝑑𝑠≀𝐢1,(4.7) where 𝐢1 is a constant independent of 𝑁.

Proof. By Ito’s formula, we obtain from (3.16) and (4.4) that 𝑑‖‖𝑒𝑁‖‖(𝑑)2+2βŸ¨π΄π‘’π‘(𝑑),𝑒𝑁2𝐹(𝑑)βŸ©π‘‘π‘‘=𝑑,𝑒𝑁(𝑑),𝑒𝑁+(𝑑)𝑁𝑗=1πœ†π‘—ξ‚πΊξ€·ξ‚€ξ‚€π‘‘,𝑒𝑁(𝑑),𝑒𝑗2𝐺𝑑𝑑+2𝑑,𝑒𝑁(𝑑),𝑒𝑁𝑑(𝑑)ξ‚ξ‚π‘Š.(4.8) Integrating (4.8) with respect to 𝑑, and using (3.13) and (3.19), we have ‖‖𝑒𝑁‖‖(𝑑)2ξ€œ+𝛼𝑑0‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)‖‖𝑒𝑑𝑠≀𝑁0β€–β€–2ξ€œ+𝐢+𝐢𝑑0‖‖𝑒𝑁‖‖(𝑠)2ξ€œπ‘‘π‘ +2𝑑0𝐺𝑠,𝑒𝑁(𝑠),𝑒𝑁𝑑(𝑠)ξ‚ξ‚π‘Š(𝑠).(4.9) Let us estimate the stochastic integral in this inequality. By Burkholder-Davis Gundy’s inequality [15], we have 𝐸sup0≀𝑠≀𝑑||||ξ€œπ‘ 0𝐺𝑠,𝑒𝑁(𝑠),𝑒𝑁𝑑(𝑠)ξ‚ξ‚π‘Š||||(𝑠)β‰€πΆπΈξ‚΅ξ€œπ‘‘0𝐺𝑠,𝑒𝑁(𝑠),𝑒𝑁(𝑠)2𝑑𝑠1/2β‰€πœ–πΈsup0≀𝑠≀𝑑‖‖𝑒𝑁‖‖(𝑠)2+πΆπœ–ξ€œπ‘‘0‖‖𝑒1+𝑁‖‖(𝑠)2𝑑𝑠,(4.10) here we have used π»Μˆπ‘œπ‘™π‘‘π‘’π‘Ÿξ…žπ‘  and Young’s inequalities; πœ– is an arbitrary positive number.
Using (4.10) and (4.9) together with appropriate choice of πœ–, we obtain
𝐸sup0≀𝑠≀𝑑‖‖𝑒𝑁‖‖(𝑠)2+2ξ‚π›ΌπΈξ€œπ‘‘0‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)𝑑𝑠≀𝐢+πΆπΈξ€œπ‘‘0‖‖𝑒𝑁‖‖(𝑠)2𝑑𝑠.(4.11) By Gronwall’s lemma, we obtain the sought estimate (4.7).

The following result is related to the higher integrability of 𝑒𝑁.

Lemma 4.2. It holds that 𝐸sup0≀𝑠≀𝑇‖‖𝑒𝑁‖‖(𝑠)π‘β‰€πΆπ‘βˆ€1≀𝑝<∞.(4.12)

Proof. By Ito’s formula, it follows from (4.4) that for 𝑝β‰₯4, we have 𝑑‖‖𝑒𝑁‖‖(𝑑)𝑝/2=𝑝2‖‖𝑒𝑁‖‖(𝑑)𝑝/2βˆ’2βŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’ξ‚¬ξ‚π΄π‘’π‘(𝑑),𝑒𝑁𝐡𝑒(𝑑)βˆ’2𝑁(𝑑),𝑒𝑁(𝑑),𝑒𝑁𝐹(𝑑)+2𝑑,𝑒𝑁(𝑑),𝑒𝑁+1(𝑑)2𝑁𝑖=1πœ†π‘–ξ‚πΊξ€·ξ‚€ξ‚€π‘‘,𝑒𝑁(𝑑),𝑒𝑖2+π‘βˆ’44𝐺𝑒𝑁(𝑑),𝑒𝑁(𝑑)2‖𝑒𝑁(𝑑)β€–2⎀βŽ₯βŽ₯βŽ₯⎦+𝑝𝑑𝑑2‖‖𝑒𝑁‖‖(𝑑)𝑝/2βˆ’2𝐺𝑑,𝑒𝑁(𝑑),𝑒𝑁𝑑(𝑑)ξ‚ξ‚π‘Š.(4.13) Using the assumptions (3.16), (3.19), (3.26), it follows that sup0≀𝑠≀𝑑‖‖𝑒𝑁‖‖(𝑠)𝑝/2≀‖‖𝑒𝑁0‖‖𝑝/2ξ€œ+𝐢𝑑0‖‖𝑒1+𝑁‖‖(𝑠)𝑝/2+𝑝𝑑𝑠2sup0≀𝑠≀𝑑||||ξ€œπ‘ 0‖‖𝑒𝑁‖‖(𝑠)𝑝/2βˆ’2𝐺𝑠,𝑒𝑁(𝑠),𝑒𝑁𝑑(𝑠)ξ‚ξ‚π‘Š||||.(4.14) Squaring the both sides of this inequality and passing to mathematical expectation, we deduce from the Martingale inequality, that is, 𝐸sup0≀𝑠≀𝑑‖‖𝑒𝑁‖‖(𝑠)𝑝‖‖𝑒≀𝐢𝑁0‖‖𝑝+𝑇+πΈξ€œπ‘‘0‖‖𝑒𝑁‖‖(𝑠)𝑝.𝑑𝑠(4.15) From Gronwall’s inequality, we deduce that 𝐸sup0≀𝑠≀𝑑‖‖𝑒𝑁‖‖(𝑠)𝑝≀𝐢𝑝(4.16) for all 1≀𝑝<∞.

We also have the following lemma.

Lemma 4.3. It holds that 𝑒𝑁 satisfies πΈξ‚΅ξ€œπ‘‡0‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)𝑑𝑠𝑝≀𝐢𝑝(4.17) for all 1≀𝑝<∞.

Proof. Using (4.9), we have ξ‚π›Όπ‘ξ‚΅ξ€œπ‘‘0‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)𝑝‖‖𝑒≀𝐢𝑁0β€–β€–2π‘ξ‚΅ξ€œ+𝐢+𝐢𝑑0‖‖𝑒𝑁‖‖(𝑠)2𝑑𝑠𝑝||||ξ€œ+𝐢𝑑0((𝐺(𝑠,𝑒𝑁(𝑠)),𝑒𝑁(𝑠)))π‘‘π‘Š||||𝑝.(4.18) Taking the mathematical expectation and use the Burkholder-Gundy’s inequality, the proof of the lemma follows from Lemma 4.2.

Lemma 4.4. It holds that 𝐸sup||πœƒ||0≀≀𝛿≀1ξ€œπ‘‡0‖‖𝑒𝑁(𝑑+πœƒ)βˆ’π‘’π‘β€–β€–(𝑑)2𝐷(𝐴)′𝑑𝑑≀𝐢𝛿.(4.19)

Proof. We note that the functions {πœ†π‘—π‘’π‘—}𝑗=1,2,… form an orthonormal basis in the dual 𝐷(𝐴)ξ…ž of 𝐷(𝐴). Let 𝑃𝑁 be the orthogonal projection of 𝐷(𝐴)ξ…ž onto the span {πœ†1𝑒1,…,πœ†π‘π‘’π‘}, that is, π‘ƒπ‘β„Ž=𝑁𝑗=1πœ†π‘—βŸ¨β„Ž,π‘’π‘—βŸ©π‘’π‘—.(4.20)
Thus (4.4) can be rewritten in an integral form as the equality between random variables with values in 𝐷(𝐴)ξ…ž as
𝑒𝑁(ξ€œπ‘‘)+𝑑0𝑃𝑁𝐴𝑒𝑁(𝐡𝑒𝑠)+𝑁(𝑠),𝑒𝑁(ξ€Έβˆ’ξ‚πΉξ€·π‘ )𝑠,𝑒𝑁(𝑠)𝑑𝑠=𝑒𝑁0+ξ€œπ‘‘0𝑃𝑁𝐺𝑠,𝑒𝑁𝑑(𝑠)π‘Š.(4.21) For any positive πœƒ, we have ‖‖𝑒𝑁(𝑑+πœƒ)βˆ’π‘’π‘β€–β€–(𝑑)𝐷(𝐴)β€²β‰€β€–β€–β€–ξ€œπ‘‘π‘‘+πœƒξ‚€ξ‚π΄π‘’π‘ξ‚π΅ξ€·π‘’(𝑠)+𝑁(𝑠),π‘’π‘ξ€Έβˆ’ξ‚πΉξ€·(𝑠)𝑠,𝑒𝑁‖‖‖(𝑠)𝑑𝑠𝐷(𝐴)β€²+β€–β€–β€–ξ€œπ‘‘π‘‘+πœƒξ‚πΊξ€·π‘ ,𝑒𝑁𝑑(𝑠)π‘Šβ€–β€–β€–π·(𝐴)β€².(4.22) Taking the square and use the properties of 𝐡𝐴, and 𝐹, we have ‖‖𝑒𝑁(𝑑+πœƒ)βˆ’π‘’π‘β€–β€–(𝑑)2𝐷(𝐴)β€²β‰€πΆπœƒ2ξ‚΅ξ€œ+𝐢𝑑𝑑+πœƒβ€–β€–π‘’π‘β€–β€–(𝑠)2𝐷(𝐴)𝑑𝑠2+𝐢sup0≀𝑑≀𝑇‖‖𝑒𝑁‖‖(𝑠)2ξ‚΅ξ€œπ‘‘π‘‘+πœƒβ€–β€–π‘’π‘β€–β€–(𝑠)𝐷(𝐴)𝑑𝑠2+πΆπœƒ2sup0≀𝑠≀𝑇‖‖𝑒𝑁‖‖(𝑠)2+β€–β€–β€–ξ€œπ‘‘π‘‘+πœƒξ‚πΊξ€·π‘ ,𝑒𝑁𝑑(𝑠)π‘Šβ€–β€–β€–2.(4.23) For fixed 𝛿, taking the supremun over πœƒβ‰€π›Ώ, integrating with respect to 𝑑, and taking the mathematical expectation, we have 𝐸sup0β‰€πœƒβ‰€π›Ώξ€œπ‘‡0‖‖𝑒𝑁(𝑑+πœƒ)βˆ’π‘’π‘β€–β€–(𝑑)2𝐷(𝐴)′𝑑𝑑≀𝐢𝛿2+πΆπΈξ€œπ‘‡0ξ‚΅ξ€œπ‘‘π‘‘+𝛿‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)𝑑𝑠2𝑑𝑑+𝐢𝐸sup0≀𝑠≀𝑇‖‖𝑒𝑁‖‖(𝑠)2ξ€œπ‘‡0ξ‚΅ξ€œπ‘‘π‘‘+𝛿‖‖𝑒𝑁‖‖(𝑠)𝐷(𝐴)𝑑𝑠2𝑑𝑑+𝐢𝛿2𝐸sup0≀𝑠≀𝑇‖‖𝑒𝑁‖‖(𝑠)2+πΈξ€œπ‘‡0sup0β‰€πœƒβ‰€π›Ώβ€–β€–β€–ξ€œπ‘‘π‘‘+πœƒξ‚πΊξ€·π‘ ,𝑒𝑁𝑑(𝑠)π‘Šβ€–β€–β€–2𝑑𝑑.(4.24) We estimate the integrals in this inequality.
We have by π»Μˆπ‘œπ‘™π‘‘π‘’π‘Ÿξ…žπ‘  inequality
𝐼1=𝐸sup0≀𝑠≀𝑇‖‖𝑒𝑁‖‖(𝑠)2ξ€œπ‘‡0ξ‚΅ξ€œπ‘‘π‘‘+𝛿‖‖𝑒𝑁‖‖(𝑠)𝐷(𝐴)𝑑𝑠2𝑑𝑑≀𝛿2𝐸sup0≀𝑠≀𝑇‖‖𝑒𝑁‖‖(𝑠)2ξ€œπ‘‡0‖‖𝑒𝑁‖‖(𝑠)2𝐷(𝐴)𝑑𝑠.(4.25) Using the π»Μˆπ‘œπ‘™π‘‘π‘’π‘Ÿξ…žπ‘  inequality and the estimates of Lemmas 4.2 and 4.3, we have 𝐼1≀𝐢𝛿2.(4.26) By Martingale’s inequality, we have 𝐼2=πΈξ€œπ‘‡0sup0β‰€πœƒβ‰€π›Ώβ€–β€–β€–ξ€œπ‘‘π‘‘+πœƒξ‚πΊξ€·π‘ ,𝑒𝑁𝑑(𝑠)π‘Šβ€–β€–β€–2β‰€π‘‘π‘‘πΈξ€œπ‘‡0ξ‚΅ξ€œπ‘‘π‘‘+𝛿‖‖𝐺(𝑠,𝑒𝑁‖‖(𝑠))2𝑑𝑠𝑑𝑑.(4.27) Using the assumptions on 𝐺 and the estimate of Lemma 4.2, we have 𝐼2≀𝐢𝛿.(4.28) Collecting the results and making a similar reasoning with πœƒ<0, we obtain from (4.24) that 𝐸sup||πœƒ||0β‰€β‰€π›Ώξ€œπ‘‡0‖‖𝑒𝑁(𝑑+πœƒ)βˆ’π‘’π‘β€–β€–(𝑑)2𝐷(𝐴)′≀𝐢𝛿(4.29)

The following lemma is from [16], and it is a compactness results which represents a variation of the compactness theorems in [17, Chapter I, Section 5]. It will be useful for us to prove the tightness property of Galerkin solution.

Proposition 4.5. For any sequences of positives reals number πœˆπ‘š,πœ‡π‘š which tend to 0 as π‘šβ†’βˆž, the injection of π‘Œπœ‡π‘›,πœˆπ‘›=ξƒ―π‘¦βˆˆπΏ2(0,𝑇;𝐷(𝐴))∩𝐿∞(0,𝑇;𝑉)∣supπ‘š1πœˆπ‘šsup||πœƒ||β‰€πœ‡π‘šξ‚΅ξ€œπ‘‡0‖𝑦(𝑑+πœƒ)βˆ’π‘¦(𝑑)β€–2𝐷(𝐴)β€²ξ‚Ά1/2ξƒ°<∞(4.30) in 𝐿2(0,𝑇;𝑉) is compact.

Furthermore π‘Œπœ‡π‘›,πœˆπ‘› is a Banach space with the norm

β€–π‘¦β€–π‘Œπœ‡π‘›π‘›,𝜈=sup0β‰€π‘‘β‰€π‘‡ξ‚΅ξ€œβ€–π‘¦(𝑑)β€–+𝑇0‖𝑦(𝑑)β€–2𝐷(𝐴)𝑑𝑑1/2+sup𝑛1πœˆπ‘›sup||πœƒ||β‰€πœ‡π‘›ξ‚΅ξ€œπ‘‡0‖𝑦(𝑑+πœƒ)βˆ’π‘¦(𝑑)β€–2𝐷(𝐴)′𝑑𝑑1/2.(4.31) Alongside with π‘Œπœ‡π‘›,πœˆπ‘›, we also consider the space 𝑋𝑝,πœ‡π‘›,πœˆπ‘›(1≀𝑝<∞) of random variables 𝑦 such that

𝐸sup0≀𝑑≀𝑇‖𝑦(𝑑)‖𝑝<∞;πΈξ‚΅ξ€œπ‘‡0‖𝑦(𝑑)β€–2𝐷(𝐴)𝑑𝑑𝑝/2<∞;𝐸sup𝑛1πœˆπ‘›sup||πœƒ||β‰€πœ‡π‘›ξ€œπ‘‡0‖𝑦(𝑑+πœƒ)βˆ’π‘¦(𝑑)β€–2𝐷(𝐴)′𝑑𝑑<∞.(4.32) Endowed with the norm

‖𝑦‖𝑋𝑛𝑛𝑝,𝜈,πœ‡=𝐸sup0≀𝑑≀𝑇‖𝑦(𝑑)‖𝑝1/𝑝+ξƒ©πΈξ‚΅ξ€œπ‘‡0‖𝑦(𝑑)β€–2𝐷(𝐴)𝑑𝑑𝑝/2ξƒͺ𝑝/2+𝐸sup𝑛1πœˆπ‘›ξƒ©sup||πœƒ||β‰€πœ‡π‘›ξ€œπ‘‡0‖𝑦(𝑑+πœƒ)βˆ’π‘¦(𝑑)β€–2𝐷(𝐴)β€²ξƒͺ𝑑𝑑1/2,(4.33)𝑋𝑝,πœ‡π‘›,πœˆπ‘› is a Banach space. The priori estimates of the preceding lemmas enable us to claim that for any 1≀𝑝<∞ and for πœ‡π‘›,πœˆπ‘› such that the series βˆ‘βˆžπ‘›=1(βˆšπœ‡π‘›/πœˆπ‘›) converges, the sequence of Galerkin solutions {π‘’π‘βˆΆπ‘βˆˆπ‘} is bounded in 𝑋𝑝,πœ‡π‘›,πœˆπ‘›.

4.1.3. Step  3. Tightness Property of Galerkin Solutions

Now, we consider the set

𝑆=𝐢(0,𝑇;π‘…π‘š)×𝐿2(0,𝑇;𝑉),(4.34) and 𝐡(𝑆) the𝜎-algebra of the Borel sets of 𝑆.

For each 𝑁, let πœ™ be the map

πœ™βˆΆΞ©βŸΆπ‘†βˆΆξ‚€πœ”βŸΌπ‘Šξ€·ξ€Έπœ”,β‹…,𝑒𝑁.πœ”,β‹…(4.35) For each 𝑁, we introduce a probability measure Π𝑁 on (𝑆,𝐡(𝑆)) by

Π𝑁(𝐴)=π‘ƒξ€·πœ™βˆ’1(𝐴)(4.36) for all 𝐴∈𝐡(𝑆). The main result of this subsection is the following.

Theorem 4.6. The family of probability measures {Π𝑁;π‘βˆˆπ‘} is tight.

Proof. For πœ€>0, we should find the compact subsets Ξ£πœ€βŠ‚πΆ(0,𝑇;π‘…π‘š),π‘Œπœ€βŠ‚πΏ2(0,𝑇;𝑉),(4.37) such that π‘ƒξ‚€πœ”βˆΆπ‘Šξ€·ξ€Έπœ”,β‹…βˆ‰Ξ£πœ€ξ‚β‰€πœ€2,(4.38)π‘ƒξ€·πœ”βˆΆπ‘’π‘ξ€·ξ€Έπœ”,β‹…βˆ‰π‘Œπœ€ξ€Έβ‰€πœ€2.(4.39) The quest for Ξ£πœ€ is made by taking account of some fact about the Wiener process such as the formula 𝐸||π‘Šξ€·π‘‘2ξ€Έβˆ’π‘Šξ€·π‘‘1ξ€Έ||2𝑗𝑑=(2π‘—βˆ’1)!2βˆ’π‘‘1𝑗,𝑗=1,2,….(4.40) For a constant πΏπœ€ depending on πœ€ to be chosen later and π‘›βˆˆπ‘, we consider the set Ξ£πœ€=ξƒ―π‘Š(β‹…)∈𝐢(0,𝑇;π‘…π‘š)∢sup𝑑1,𝑑2||π‘‘βˆˆ[0,𝑇],2βˆ’π‘‘1||≀1/𝑛6𝑛||π‘Šξ€·π‘‘2ξ€Έξ€·π‘‘βˆ’π‘Š1ξ€Έ||β‰€πΏπœ€ξƒ°(4.41) Making use of Markov’s inequality: 1𝑃(πœ”βˆΆπœ‰(πœ”)β‰₯𝛼)β‰€π›Όπ‘˜πΈξ‚ƒ||||πœ‰(πœ”)π‘˜ξ‚„(4.42) for a random variable πœ‰ on (Ξ©,𝐹,𝑃) and positives variables 𝛼 and π‘˜, we get π‘ƒξ‚€πœ”βˆΆπ‘Šξ€·ξ€Έπœ”,β‹…βˆ‰Ξ£πœ€ξ‚β‰€π‘ƒξƒ¬ξšπ‘›ξƒ―πœ”βˆΆsup𝑑1,𝑑2∈[]∢||𝑑0,𝑇2βˆ’π‘‘1||<1/𝑛6||π‘Šξ€·π‘‘2ξ€Έβˆ’π‘Šξ€·π‘‘1ξ€Έ||>πΏπœ€π‘›β‰€ξƒ°ξƒ­βˆžξ“π‘›π‘›=16βˆ’1𝑖=0ξ‚΅π‘›πΏπœ€ξ‚Ά4𝐸sup𝑖𝑇/𝑛6≀𝑑≀(𝑖+1)𝑇/𝑛6||π‘Š(𝑑)βˆ’π‘Šξ€·π‘–π‘‡π‘›βˆ’6ξ€Έ||4β‰€π‘βˆžξ“π‘›=1ξ‚΅π‘›πΏπœ€ξ‚Ά4ξ€·π‘‡π‘›βˆ’6ξ€Έ2𝑛6=𝑐𝐿4πœ€βˆžξ“π‘›=11𝑛2,(4.43) we choose 𝐿4πœ€=2πΆπœ€βˆžβˆ’1𝑛=11𝑛2(4.44) to get (4.38).
Next we choose π‘Œπœ€ as a ball of radius π‘€πœ€ in π‘Œπœ‡π‘›,πœˆπ‘› centered at zero and with πœ‡π‘›,πœˆπ‘›, independent of πœ€, converging to zero, and such that βˆ‘π‘›(βˆšπœ‡π‘›/πœˆπ‘›) converges.
From Proposition 4.5, π‘Œπœ€ is a compact subset of 𝐿2(0,𝑇;𝑉).
We have further
π‘ƒξ€·πœ”βˆΆπ‘’π‘(πœ”,β‹…)βˆ‰π‘Œπœ€ξ€Έβ‰€π‘ƒξ‚€β€–β€–π‘’πœ”βˆΆπ‘β€–β€–π‘Œπœ‡π‘›π‘›,𝜈>π‘€πœ€ξ‚β‰€1π‘€πœ€πΈβ€–β€–π‘’π‘β€–β€–π‘Œπœ‡π‘›π‘›,πœˆβ‰€π‘π‘€πœ€,(4.45) choosing π‘€πœ€=2π‘πœ€βˆ’1, we get (4.39).
From (4.38) and (4.39), we have
π‘ƒξ‚€πœ”βˆΆπ‘Š(πœ”,β‹…)βˆˆΞ£πœ€;𝑒𝑁(πœ”,β‹…)βˆˆπ‘Œπœ€ξ‚β‰₯1βˆ’πœ€,(4.46) this proves that Ξ π‘ξ€·Ξ£πœ€Γ—π‘Œπœ€ξ€Έβ‰₯1βˆ’πœ€,βˆ€π‘βˆˆβ„•.(4.47)

4.1.4. Step  4. Applications of Prokhorov and Skorokhod Results

From the tightness property of {Π𝑁} and Prokhorov’s theorem [12], we have that there exist a subsequence {Π𝑁𝑗} and a measure Ξ  such that Π𝑁𝑗→Π weakly.

By Skorokhod’s theorem [13], there exist a probability space (Ξ©,β„±,𝑃) and random variables (π‘Šπ‘π‘—,𝑒𝑁𝑗),(π‘Š,𝑒) on (Ξ©,β„±,𝑃) with values in 𝑆 such that

thelawofξ‚€π‘Šπ‘π‘—,𝑒𝑁𝑗isΠ𝑁𝑗,(4.48)thelawof(π‘Š,𝑒)isΞ ,(4.49)ξ‚€π‘Šπ‘π‘—,π‘’π‘π‘—ξ‚βŸΆ(π‘Š,𝑒)in`𝑆,𝑃-a.s.(4.50) Hence, {π‘Šπ‘π‘—} is a sequence of an π‘š-dimensional standard Wiener process.

Let ℱ𝑑=𝜎{π‘Š(𝑠),𝑒(𝑠),0≀𝑠≀𝑑}.

Arguing as in [16], we prove that π‘Š(𝑑) is an π‘š-dimensional ℱ𝑑 standard Wiener process and the pair (π‘Šπ‘π‘—,𝑒𝑁𝑗) satisfies the equation

𝑒𝑁𝑗(ξ€œπ‘‘)+πœˆπ‘‘0𝑃𝑁𝑗𝐴𝑒𝑁𝑗(ξ€œπ‘ )𝑑𝑠+𝑑0𝑃𝑁𝑗𝐡𝑒𝑁𝑗(𝑠),𝑒𝑁𝑗(ξ€Έ=ξ€œπ‘ )𝑑𝑠𝑑0𝑃𝑁𝑗𝐹𝑠,π‘’π‘π‘—ξ€Έξ€œ(𝑠)𝑑𝑠+𝑑0𝑃𝑁𝑗𝐺𝑠,𝑒𝑁𝑗(𝑠)π‘‘π‘Šπ‘π‘—+𝑒𝑁𝑗0.(4.51)

4.1.5. Step  5. Passage to the Limit

From (4.51), it follows that 𝑒𝑁𝑗 satisfies the results of the Lemmas 4.2, 4.3, and 4.4. Therefore, we have for 𝑝β‰₯1 the a priori estimates

𝐸sup0≀𝑑≀𝑇‖‖𝑒𝑁𝑗(𝑑)β€–β€–π‘πΈξ‚΅ξ€œβ‰€πΆ;𝑇0‖‖𝑒𝑁𝑗‖‖(𝑑)2𝐷(𝐴)𝑑𝑑𝑝≀𝐢;𝐸sup0β‰€πœƒβ‰€π›Ώξ€œπ‘‡0‖‖𝑒𝑁𝑗(𝑑+πœƒ)βˆ’π‘’π‘π‘—β€–β€–2𝐷(𝐴)′𝑑𝑑≀𝐢(𝛼)𝛿(4.52) thus modulo the extraction of a subsequence denoted again by 𝑒𝑁𝑗, we have

π‘’π‘π‘—βŸΆπ‘’weaklyβˆ—in𝐿𝑝(Ξ©,β„±,𝑃;πΏβˆžπ‘’(0,𝑇;𝑉));π‘π‘—βŸΆπ‘’weaklyin𝐿𝑝Ω,β„±,𝑃;𝐿2ξ€Έ;(0,𝑇;𝐷(𝐴))𝐸sup0≀𝑑≀𝑇‖𝑒(𝑑)β€–π‘ξ‚€βˆ«β‰€πΆ;𝐸𝑇0‖𝑒(𝑑)β€–2𝐷(𝐴)𝑑𝑑𝑝≀𝐢;𝐸sup0β‰€πœƒβ‰€π›Ώβˆ«π‘‡0‖𝑒(𝑑+πœƒ)βˆ’π‘’(𝑑)β€–2𝐷(𝐴)′𝑑𝑑≀𝐢𝛿.(4.53) By (4.50) and the first estimate in (4.52) and Vitali’s theorem, we have

π‘’π‘π‘—βŸΆπ‘’stronglyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€Έ,(0,𝑇,𝑉)(4.54) and thus modulo the extraction of a subsequence and for almost every (πœ”,𝑑) with respect to the measure π‘‘π‘ƒβŠ—π‘‘π‘‘:

π‘’π‘π‘—βŸΆπ‘’in𝑉.(4.55) This convergence together with the condition on 𝐹, the first estimate in (4.52) and Vitali’s theorem, give

𝐹⋅,π‘’π‘π‘—ξ€ΈβŸΆξ‚πΉ(β‹…)(β‹…,𝑒(β‹…))stronglyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€Έ,ξ€œ(0,𝑇,𝑉)𝑑0𝐹𝑠,𝑒𝑁𝑗(𝑠)ξ€Έξ€œπ‘‘π‘ βŸΆπ‘‘0𝐹(𝑠,𝑒(𝑠))𝑑𝑠stronglyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€Έ.(0,𝑇,𝑉)(4.56) As

π‘’π‘π‘—βŸΆπ‘’weaklyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€Έ,(0,𝑇;𝐷(𝐴))(4.57) then

ξ€œπ‘‘0𝐴𝑒𝑁𝑗(ξ€œπ‘ )π‘‘π‘ βŸΆπ‘‘0𝐴𝑒(𝑠)𝑑𝑠weaklyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€·0,𝑇;𝐷(𝐴)ξ…ž.ξ€Έξ€Έ(4.58) We also have

ξ€œπ‘‘0𝐡𝑒𝑁𝑗(𝑠),𝑒𝑁𝑗(ξ€Έξ€œπ‘ )π‘‘π‘ βŸΆπ‘‘0𝐡(𝑒(𝑠),𝑒(𝑠))𝑑𝑠weaklyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€·0,𝑇;𝐷(𝐴)ξ…ž.ξ€Έξ€Έ(4.59) In fact, since 𝐿∞(Ω×(0,𝑇),𝑑𝑃×𝑑𝑑;𝐷(𝐴)) is dense in 𝐿2(Ξ©,β„±,𝑃;𝐿2(0,𝑇;𝐷(𝐴))), and 𝐡(𝑒𝑁𝑗(𝑠),𝑒𝑁𝑗(𝑠)) is bounded in 𝐿2(Ξ©,β„±,𝑃;𝐿2(0,𝑇;𝐷(𝐴)ξ…ž)) it suffices to prove that forallπœ‘βˆˆπΏβˆž(Ω×(0,𝑇),𝑑𝑃×𝑑𝑑;𝐷(𝐴)),

πΈξ€œπ‘‡0βŸ¨ξ‚π΅ξ€·π‘’π‘π‘—(𝑠),𝑒𝑁𝑗(𝑠),πœ‘(𝑠)⟩𝐷(𝐴)β€²ξ€œπ‘‘π‘ βŸΆπΈπ‘‡0βŸ¨ξ‚π΅(𝑒(𝑠),𝑒(𝑠)),πœ‘(𝑠)⟩𝐷(𝐴)′𝑑𝑠.(4.60) Indeed, we have

πΈξ€œπ‘‡0βŸ¨ξ‚π΅ξ€·π‘’π‘π‘—(𝑠),π‘’π‘π‘—ξ€Έβˆ’ξ‚(𝑠)𝐡(𝑒(𝑠),𝑒(𝑠)),πœ‘(𝑠)⟩𝐷(𝐴)β€²ξ€œπ‘‘π‘ =𝐸𝑇0𝐡(𝑒𝑁𝑗(𝑠)βˆ’π‘’(𝑠),𝑒𝑁𝑗(𝑠)),πœ‘(𝑠)𝐷(𝐴)β€²ξ€œπ‘‘π‘ +𝐸𝑇0𝐡(𝑒(𝑠),𝑒𝑁𝑗(𝑠)βˆ’π‘’(𝑠)),πœ‘(𝑠)𝐷(𝐴)′𝑑𝑠=𝐼1𝑗+𝐼2𝑗,𝐼1π‘—ξ€œ=𝐸𝑇0𝐡𝑒𝑁𝑗(𝑠)βˆ’π‘’(𝑠),𝑒𝑁𝑗(𝑠),πœ‘(𝑠)𝐷(𝐴)′𝑑𝑠(4.61) By the property (3.17) of 𝐡, we have

𝐼1π‘—ξ€œβ‰€πΆπΈπ‘‡0‖‖𝑒𝑁𝑗‖‖‖‖𝑒(𝑠)βˆ’π‘’(𝑠)𝑁𝑗‖‖(𝑠)𝐷(𝐴)||||π΄πœ‘(𝑠)𝑑𝑠,(4.62) applying Cauchy-Schwarz inequality

𝐼1π‘—β‰€πΆπœ‘ξ‚΅πΈξ€œπ‘‡0‖‖𝑒𝑁𝑗‖‖(𝑠)βˆ’π‘’(𝑠)2𝑑𝑠1/2ξ‚΅πΈξ€œπ‘‡0‖‖𝑒𝑁𝑗‖‖(𝑠)2𝐷(𝐴)𝑑𝑠1/2.(4.63) Since

π‘’π‘π‘—βŸΆπ‘’stronglyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€Έ,(0,𝑇;𝑉)(4.64) and 𝑒𝑁𝑗 is bounded in 𝐿2(Ξ©,β„±,𝑃;𝐿2(0,𝑇;𝐷(𝐴))), we conclude that

𝐼1π‘—βŸΆ0asπΌπ‘—βŸΆβˆž.2π‘—ξ€œ=𝐸𝑇0βŸ¨ξ‚π΅ξ€·π‘’(𝑠),π‘’π‘π‘—ξ€Έβˆ’π‘’(𝑠),πœ‘(𝑠)⟩𝐷(𝐴)′𝑑𝑠.(4.65) Again thanks to the property (3.18) of 𝐡, as

π‘’π‘π‘—βŸΆπ‘’weaklyin𝐿2ξ€·Ξ©,β„±,𝑃;𝐿2ξ€Έ,(0,𝑇;𝐷(𝐴))(4.66) we obtain 𝐼2𝑗→0 as π‘—β†’βˆž since any strongly continuous linear operator is weakly continuous. We are now left with the proof of

ξ€œπ‘‘0𝐺𝑠,𝑒𝑁𝑗(𝑠)π‘‘π‘Šπ‘π‘—(ξ€œπ‘ )βŸΆπ‘‘0𝐺(𝑠,𝑒(𝑠))π‘‘π‘Š(𝑠)weaklyβˆ—πΏ2ξ€·Ξ©,β„±,𝑃;πΏβˆžξ€·0,𝑇;𝐷(𝐴)ξ…ž,ξ€Έξ€Έ(4.67) which can be prove with the same argument like in [16].

Collecting all the convergence results, we deduce that

ξ€œπ‘’(𝑑)+πœˆπ‘‘0ξ‚ξ€œπ΄π‘’(𝑠)𝑑𝑠+𝑑0=ξ€œπ΅(𝑒(𝑠),𝑒(𝑠))𝑑𝑠𝑑0ξ‚ξ€œπΉ(𝑠,𝑒(𝑠))𝑑𝑠+𝑑0𝐺(𝑠,𝑒(𝑠))π‘‘π‘Š(𝑠)+𝑒0,𝑃-a.s.(4.68) as the equality in 𝐷(𝐴)ξ…ž.

We have 𝐡(𝑒,𝑒)∈𝐿2(Ξ©,β„±,𝑃;𝐿∞(0,𝑇;𝐷(𝐴)ξ…ž)), ξ‚ξ‚π΄π‘’βˆ’πΉ(𝑑,𝑒)∈𝐿2(Ξ©,β„±,𝑃;𝐿∞(0,𝑇;𝐷(𝐴)ξ…ž)), 𝐺(𝑑,𝑒)∈𝐿2(Ξ©,β„±,𝑃;𝐿∞(0,𝑇;π‘‰βŠ—π‘š)).

Thus, from the classical results in [18] (see also [19]), we deduce from (4.68) that 𝑒 is 𝑃-a.s. continuous with values in 𝑉.

4.1.6. Step  6. Existence of the Pressure

For the existence of the pressure, we use a generalization of the Rham’s theorem processes [20, Theorem 4.1, Remark 4.3]. From (3.6), we have for all π‘£βˆˆπ’±,

ξ‚¬βˆ’πœ•π‘‘(π‘’βˆ’π›ΌΞ”π‘’)βˆ’πœˆ(π΄π‘’βˆ’π›ΌΞ”(𝐴𝑒))βˆ’(π‘’β‹…βˆ‡)(π‘’βˆ’π›ΌΞ”π‘’)+π›Όβˆ‡π‘’βˆ—β‹…Ξ”π‘’+𝐹(β‹…,𝑒)+𝐺(β‹…,𝑒)π‘‘π‘Šξ‚­π‘‘π‘‘,π‘£ξ€·π’Ÿβ€²ξ€Έ(𝐷)3Γ—(π’Ÿ(𝐷))3=0.(4.69) We denote

β„Ž=βˆ’πœ•π‘‘(π‘’βˆ’π›ΌΞ”π‘’)βˆ’πœˆ(π΄π‘’βˆ’π›ΌΞ”(𝐴𝑒))βˆ’(π‘’β‹…βˆ‡)(π‘’βˆ’π›ΌΞ”π‘’)+π›Όβˆ‡π‘’βˆ—β‹…Ξ”π‘’+𝐹(β‹…,𝑒)+𝐺(β‹…,𝑒)π‘‘π‘Š.𝑑𝑑(4.70) We will prove that the regularity on 𝑒, implies that

β„ŽβˆˆπΏ2ξ‚€Ξ©,ℱ𝑑,𝑃;π»βˆ’1𝐻0,𝑑;βˆ’2ξ€Έ(𝐷)3.(4.71) By (2.7) and (2.9), we have as π‘’βˆˆπΏ4(Ξ©,β„±,𝑃;𝐿2(0,𝑇;𝐷(𝐴))),

(π‘’β‹…βˆ‡(π‘’βˆ’π›ΌΞ”π‘’))+βˆ‡π‘’βˆ—β‹…Ξ”π‘’βˆˆπΏ2ξ‚€Ξ©,ℱ𝑑,𝑃;𝐿1𝐻0,𝑑;βˆ’1ξ€Έ(𝐷)3,ξ‚ξ‚π΄π‘’βˆ’π›ΌΞ”(𝐴𝑒)∈𝐿4ξ‚€Ξ©,ℱ𝑑,𝑃;𝐿2𝐻0,𝑑;βˆ’2ξ€Έ(𝐷)3.(4.72) We also have

π‘’βˆ’π›ΌΞ”π‘’βˆˆπΏ4ξ‚€Ξ©,ℱ𝑑,𝑃;𝐿2𝐿0,𝑑;2ξ€Έ(𝐷)3,πœ•ξ‚ξ‚π‘‘(π‘’βˆ’π›ΌΞ”π‘’)∈𝐿4ξ‚€Ξ©,ℱ𝑑,𝑃;π»βˆ’1𝐿0,𝑑;2ξ€Έ(𝐷)3[].,βˆ€π‘‘βˆˆ0,𝑇(4.73) Again, as π‘’βˆˆπΏ4(Ξ©,β„±,𝑃;𝐢([0,𝑇];𝑉)), then its follows that

𝐹(𝑑,𝑒)∈𝐿4ξ‚€Ξ©,ℱ𝑑,𝑃;𝐿2𝐻0,𝑑;βˆ’1ξ€Έ(𝐷)3,𝐺(𝑑,𝑒)π‘‘π‘Šπ‘‘π‘‘βˆˆπΏ4ξ‚€Ξ©,ℱ𝑑,𝑃;π‘Šβˆ’1,βˆžξ‚€ξ€·πΏ0,𝑑;2ξ€Έ(𝐷)3,(4.74) for all π‘‘βˆˆ[0,𝑇].

Then β„ŽβˆˆπΏ2(Ξ©,ℱ𝑑,𝑃;π»βˆ’1(0,𝑑;(π»βˆ’2(𝐷))3), and

βŸ¨β„Ž,π‘£βŸ©ξ€·π’Ÿβ€²ξ€Έ(𝐷)3Γ—(π’Ÿ(𝐷))3=0,βˆ€π‘£βˆˆπ’±.(4.75) Therefore, by a generalization of the Rham theorem processes [20], there exists a unique Μƒπ‘βˆˆπΏ2(Ξ©,ℱ𝑑,𝑃;π»βˆ’1(0,𝑑;(π»βˆ’1(𝐷))3)such that𝑃-a.s.

ξ€œβˆ‡Μƒπ‘=β„Ž,𝐷̃𝑝𝑑π‘₯=0,thatis,(3.7).(4.76) Theorem 3.2 is proved.

4.2. Proof of Corollary 3.3

Proof. We will prove the pathwise uniqueness which implies uniqueness of weak solutions. Let 𝐿𝐹 and 𝐿𝐺 be two real such that ‖𝐹(𝑑,𝑒)βˆ’πΉ(𝑑,𝑣)β€–(π»βˆ’1(𝐷))3β‰€πΏπΉβ€–π‘’βˆ’π‘£β€–,‖𝐺(𝑑,𝑒)βˆ’πΊ(𝑑,𝑣)β€–((𝐿2(𝐷))3)π‘šβ‰€πΏπΊβ€–π‘’βˆ’π‘£β€–.(4.77) Then 𝐹 and 𝐺 are defined, respectively, by (3.14) and (3.20) satisfying ‖‖‖‖𝐹(𝑑,𝑒)βˆ’πΉ(𝑑,𝑣)π‘‰ξ‚πΉβ€–β€–ξ‚ξ‚β€–β€–β‰€πΏβ€–π‘’βˆ’π‘£β€–,𝐺(𝑑,𝑒)βˆ’πΊ(𝑑,𝑒)π‘‰βŠ—π‘šξ‚πΊβ€–β‰€πΏπ‘’βˆ’π‘£β€–.(4.78) Let 𝑒1 and 𝑒2 two weak solutions of problem (1.1) defined on the same probability space together with the same Wiener process and starting from the same initial value 𝑒0.
We denote 𝑒=𝑒1βˆ’π‘’2. Take πœ‡>0 to be defined later and ∫𝜌(𝑑)=exp(βˆ’πœ‡<