Boundary Value Problems
Volume 2010 (2010), Article ID 723018, 31 pages
doi:10.1155/2010/723018
Research Article

On the Strong Solution for the 3D Stochastic Leray-Alpha Model

Gabriel Deugoue1,2 and Mamadou Sango1

1Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
2Department of Mathematics and Computer Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon

Received 13 August 2009; Accepted 27 January 2010

Academic Editor: Vicentiu D. Radulescu

Copyright © 2010 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 prove the existence and uniqueness of strong solution to the stochastic Leray- 𝛼 equations under appropriate conditions on the data. This is achieved by means of the Galerkin approximation scheme. We also study the asymptotic behaviour of the strong solution as alpha goes to zero. We show that a sequence of strong solutions converges in appropriate topologies to weak solutions of the 3D stochastic Navier-Stokes equations.

1. Introduction

It is computationally expensive to perform reliable direct numerical simulation of the Navier- Stokes equations for high Reynolds number flows due to the wide range of scales of motion that need to be resolved. The use of numerical models allows researchers to simulate turbulent flows using smaller computational resources. In this paper, we study a particular subgrid-scale turbulence model known as the Leray-alpha model (Leray- 𝛼 ).

We are interested in the study of the probabilistic strong solutions of the 3D Leray-alpha equations, subject to space periodic boundary conditions, in the case in which random perturbations appear. To be more precise, let 𝒯 = [ 0 , 𝐿 ] 3 , 𝑇 > 0 , and consider the system 𝑑 𝑢 − 𝛼 2 + Δ 𝑢 − 𝜈 Δ 𝑢 − 𝛼 2 Δ 𝑢 − 𝑢 ⋅ ∇ 𝑢 − 𝛼 2 Δ 𝑢 + ∇ 𝑝 𝑑 𝑡 = 𝐹 ( 𝑡 , 𝑢 ) 𝑑 𝑡 + 𝐺 ( 𝑡 , 𝑢 ) 𝑑 𝑊 i n ( 0 , 𝑇 ) × 𝒯 , ∇ ⋅ 𝑢 = 0 i n ( 0 , 𝑇 ) × 𝒯 , 𝑢 ( 𝑡 , 𝑥 ) i s p e r i o d i c i n 𝑥 , 𝒯 𝑢 𝑑 𝑥 = 0 , 𝑢 ( 0 ) = 𝑢 0 i n 𝒯 , ( 1 . 1 ) where 𝑢 = ( 𝑢 1 , 𝑢 2 , 𝑢 3 ) and 𝑝 are unknown random fields on [ 0 , 𝑇 ] × 𝒯 , representing, respectively, the velocity and the pressure, at each point of [ 0 , 𝑇 ] × 𝒯 , of an incompressible viscous fluid with constant density filling the domain 𝒯 . The constant 𝜈 > 0 and 𝛼 represent, respectively, the kinematic viscosity of the fluid and spatial scale at which fluid motion is filtered. The terms 𝐹 ( 𝑡 , 𝑢 ) and 𝐺 ( 𝑡 , 𝑢 ) 𝑑 𝑊 are external forces depending eventually on 𝑢 , where 𝑊 is an 𝑅 𝑚 -valued standard Wiener process. Finally, 𝑢 0 is a given random initial velocity field.

The deterministic version of (1.1), that is, when 𝐺 = 0 , has been the object of intense investigation over the last years. The initial motivation was to find a closure model for the 3D turbulence averaged Reynolds number; for more details, we refer to [1] and the references therein. A key interest in the model is the fact that it serves as a good approximation of the 3D Navier-Stokes equations. It is readily seen that when 𝛼 = 0 , the problem reduces to the usual 3D Navier-Stokes equations. Many important results have been obtained in the deterministic case. More precisely, the global wellposedness of weak solutions for the deterministic Leray-alpha equations has been established in [2] and also their relation with Navier-Stokes equations as 𝛼 approaches zero. The global attractor was constructed in [1, 3].

The addition of white noise driven terms to the basic governing equations for a physical system is natural for both practical and theoretical applications. For example, these stochastically forced terms can be used to account for numerical and empirical uncertainties and thus provide a means to study the robustness of a basic model. Specifically in the context of fluids, complex phenomena related to turbulence may also be produced by stochastic perturbations. For instance, in the recent work of Mikulevicius and Rozovskii [4], such terms are shown to arise from basic physical principals. To the best of our knowledge, there is no systematic work for the 3D stochastic Leray- 𝛼 model.

In this paper, we will prove the existence and uniqueness of strong solutions to our stochastic Leray- 𝛼 equations under appropriate conditions on the data, by approximating it by means of the Galerkin method (see Theorem 2.3). Here, the word “strong” means “strong” in the sense of the theory of stochastic differential equations, assuming that the stochastic processes are defined on a complete probability space and the Wiener process is given in advance. Since we consider the strong solution of the stochastic Leray-alpha equations, we do not need to use the techniques considered in the case of weak solutions (see [5–9]). The techniques applied in this paper use in particular the properties of stopping times and some basic convergence principles from functional analysis (see [10–13]). An important result, which cannot be proved in the case of weak solutions, is that the Galerkin approximations converge in mean square to the solution of the stochastic Leray-alpha equations (see Theorem 2.4). We can prove by using the property of higher-order moments for the solution. Moreover, as in the deterministic case [2], we take limits 𝛼 → 0 . We study the behavior of strong solutions as 𝛼 approaches 0 . More precisely, we show that, under this limit, a subsequence of solutions in question converges to a probabilistic weak solutions for the 3D stochastic Navier-Stokes equations (see Theorem 6.5). This is reminiscent of the vanishing viscosity method; see, for instance, [14, 15].

This paper is organized as follows. In Section 2, we formulate the problem and state the first result on the existence and uniqueness of strong solutions for the 3D stochastic Leray- 𝛼 model. In Section 3, we introduce the Galerkin approximation of our problem and derive crucial a priori estimates for its solutions. Section 4 is devoted to the proof of the existence and uniqueness of strong solutions for the 3D stochastic Leray- 𝛼 model. In Section 5, We prove the convergence result of Theorem 2.4. In Section 6, we study the asymptotic behavior of the strong solutions for the 3D stochastic Leray- 𝛼 model as 𝛼 approaches 0 .

2. Statement of the Problem and the First Main Result

Let 𝒯 = [ 0 , 𝐿 ] 3 . We denote by 𝐶 ∞ p e r ( 𝒯 ) 3 the space of all 𝒯 -periodic 𝐶 ∞ vector fields defined on 𝒯 . We set

𝒱 = Φ ∈ 𝐶 ∞ p e r ( 𝒯 ) 3 / 𝒯 Φ 𝑑 𝑥 = 0 ; ∇ ⋅ Φ = 0 . ( 2 . 1 ) We denote by 𝐻 and 𝑉 the closure of the set 𝒱 in the spaces 𝐿 2 ( 𝒯 ) 3 and 𝐻 1 ( 𝒯 ) 3 , respectively. Then 𝐻 is a Hilbert space equipped with the inner product of 𝐿 2 ( 𝒯 ) 3 . 𝑉 is Hilbert space equipped with inner product of 𝐻 1 ( 𝒯 ) 3 . We denote by ( ⋅ , ⋅ ) and | ⋅ | the inner product and norm in 𝐻 . The inner product and norm in 𝑉 are denoted by ( ( ⋅ , ⋅ ) ) and ‖ ⋅ ‖ , respectively. Let 𝐴 = − 𝒫 Δ be the Stokes operator with domain 𝐷 ( 𝐴 ) = 𝐻 2 ( 𝒯 ) 3 ∩ 𝑉 , where 𝒫 ∶ 𝐿 2 ( 𝒯 ) 3 → 𝐻 is the Leray projector. 𝐴 is an isomorphism from 𝑉 to 𝑉 (the dual space of 𝑉 ) with compact inverse, hence 𝐴 has eigenvalues { 𝜆 𝑘 } ∞ 𝑘 = 1 , t h a t i s , 4 𝜋 2 / 𝐿 2 = 𝜆 1 ≤ 𝜆 2 ≤ ⋯ ≤ 𝜆 𝑛 → ∞ ( 𝑛 → ∞ ) and corresponding eigenfunctions { 𝑤 𝑘 } ∞ 𝑘 = 1 which form an orthonormal basis of 𝐻 such that 𝐴 𝑤 𝑘 = 𝜆 𝑘 𝑤 𝑘 .

We also have

⟨ 𝐴 𝑣 , 𝑣 ⟩ 𝑉 ′ ≥ 𝛽 ‖ 𝑣 ‖ 2 ( 2 . 2 ) for all 𝑣 ∈ 𝑉 , where 𝛽 > 0 and ⟨ ⋅ , ⋅ ⟩ 𝑉 ′ denotes the duality between 𝑉 and 𝑉 .

Following the notations common in the study of Navier-Stokes equations, we set

𝐵 ( 𝑢 , 𝑣 ) = 𝒫 ( 𝑢 ⋅ ∇ ) 𝑣 ∀ 𝑢 , 𝑣 ∈ 𝑉 . ( 2 . 3 ) Then (see [16–18])

⟨ 𝐵 ( 𝑢 , 𝑣 ) , 𝑣 ⟩ 𝑉 ′ = 0 ∀ 𝑢 , 𝑣 ∈ 𝑉 , ( 2 . 4 ) ⟨ 𝐵 ( 𝑢 , 𝑣 ) , 𝑤 ⟩ 𝑉 ′ = − ⟨ 𝐵 ( 𝑢 , 𝑤 ) , 𝑣 ⟩ 𝑉 ′ | | | | | | | | | | ∀ 𝑢 , 𝑣 , 𝑤 ∈ 𝑉 , ( 2 . 5 ) ( 𝐵 ( 𝑢 , 𝑣 ) , 𝑤 ) ≤ 𝐶 𝐴 𝑢 ‖ 𝑣 ‖ | 𝑤 | , ∀ 𝑢 ∈ 𝐷 ( 𝐴 ) , 𝑣 ∈ 𝑉 , 𝑤 ∈ 𝐻 , ( 2 . 6 ) ⟨ 𝐵 ( 𝑢 , 𝑣 ) , 𝑤 ⟩ 𝐷 ( 𝐴 ) ′ | | | | | | | | ≤ 𝐶 | 𝑢 | ‖ 𝑣 ‖ 𝐴 𝑤 , ∀ 𝑢 ∈ 𝐻 , 𝑣 ∈ 𝑉 , 𝑤 ∈ 𝐷 ( 𝐴 ) , ( 2 . 7 ) ⟨ 𝐵 ( 𝑢 , 𝑣 ) , 𝑤 ⟩ 𝑉 ′ | | ≤ 𝐶 | 𝑢 | 1 / 4 ‖ 𝑢 ‖ 3 / 4 | 𝑣 | 1 / 4 ‖ 𝑣 ‖ 3 / 4 ‖ | | | | 𝑤 ‖ , ∀ 𝑢 ∈ 𝑉 , 𝑣 ∈ 𝑉 , 𝑤 ∈ 𝑉 , ( 2 . 8 ) ( 𝐵 ( 𝑢 , 𝑣 ) , 𝑤 ) ≤ 𝐶 | 𝑢 | 1 / 4 ‖ 𝑢 ‖ 3 / 4 ‖ 𝑣 ‖ 1 / 4 | | | | 𝐴 𝑣 3 / 4 | 𝑤 | , ∀ 𝑢 ∈ 𝑉 , 𝑣 ∈ 𝐷 ( 𝐴 ) , 𝑤 ∈ 𝐻 . ( 2 . 9 )

Let ( Ω , ℱ , 𝑃 ) be a complete probability space and { ℱ 𝑡 } 0 ≤ 𝑡 ≤ 𝑇 an increasing and right-continuous family of sub- 𝜎 -algebras of ℱ such that ℱ 0 contains all the P-null sets of ℱ . Let 𝑊 be a 𝑅 𝑚 -valued Wiener process on ( Ω , ℱ , { ℱ 𝑡 } 0 ≤ 𝑡 ≤ 𝑇 , 𝑃 ) .

We now introduce some probabilistic evolution spaces.

Let 𝑋 be a Banach space. For 𝑟 , 𝑝 ≥ 1 , we denote by

𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 𝑟 ( 0 , 𝑇 ; 𝑋 ) ) ( 2 . 1 0 ) the space of functions 𝑢 = 𝑢 ( 𝑥 , 𝑡 , 𝜔 ) with values in 𝑋 defined on [ 0 , 𝑇 ] × Ω and such that

( 1 ) 𝑢 is measurable with respect to ( 𝑡 , 𝜔 ) and for each 𝑡 , 𝑢 is ℱ 𝑡 measurable,( 2 ) 𝑢 ∈ 𝑋 for almost all ( 𝑡 , 𝜔 ) and

‖ 𝑢 ‖ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 𝑟 ( 0 , 𝑇 ; 𝑋 ) ) = 𝐸 𝑇 0 ‖ 𝑢 ‖ 𝑟 𝑋 𝑑 𝑡 𝑝 / 𝑟 1 / 𝑟 < ∞ , ( 2 . 1 1 ) where 𝐸 denote the mathematical expectation with respect to the probability measure 𝑃 .

The space 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 𝑟 ( 0 , 𝑇 ; 𝑋 ) ) so defined is a Banach space.

When 𝑟 = ∞ , the norm in 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝑋 ) ) is given by

‖ 𝑢 ‖ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝑋 ) ) = 𝐸 s u p 0 ≤ 𝑡 ≤ 𝑇 ‖ 𝑢 ‖ 𝑝 𝑋 1 / 𝑝 . ( 2 . 1 2 )

We make precise our assumptions on 𝐹 and 𝐺 . We suppose that 𝐹 and 𝐺 are measurable Lipschitz mappings from Ω × ( 0 , 𝑇 ) × 𝐻 into 𝐻 and from Ω × ( 0 , 𝑇 ) × 𝐻 i n t o 𝐻 ⊗ 𝑚 , respectively. More exactly, assume that, for all 𝑢 , 𝑣 ∈ 𝐻 , 𝐹 ( ⋅ , 𝑢 ) and 𝐺 ( ⋅ , 𝑢 ) are ℱ 𝑡 -adapted, and 𝑑 𝑃 × 𝑑 𝑡 − a . e . i n Ω × ( 0 , 𝑇 )

| | | | 𝐹 ( 𝑡 , 𝑢 ) − 𝐹 ( 𝑡 , 𝑣 ) 𝐻 ≤ 𝐿 𝐹 | | | | | 𝑢 − 𝑣 | , 𝐹 ( 𝑡 , 0 ) = 0 , 𝐺 ( 𝑡 , 𝑢 ) − 𝐺 ( 𝑡 , 𝑣 ) 𝐻 ⊗ 𝑚 ≤ 𝐿 𝐺 𝐺 | 𝑢 − 𝑣 | , ( 𝑡 , 0 ) = 0 . ( 2 . 1 3 ) Here 𝐻 ⊗ 𝑚 is the product of 𝑚 copies of 𝐻 .

Finally, we assume that 𝑢 0 ∈ 𝐿 2 ( Ω , ℱ 0 , 𝑃 ; 𝐷 ( 𝐴 ) ) .

Remark. The condition 10 is given only to simplify the calculations. It can be omitted; in which case one could use the estimate | | | | 𝐹 ( 𝑡 , 𝑢 ) 2 ≤ 2 𝐿 2 𝐹 | 𝑢 | 2 | | | | + 2 𝐹 ( 𝑡 , 0 ) 2 ( 2 . 1 4 ) that follows from the Lipschitz condition. The same remark applies to 𝐺 .

Alongside problem (1.1), we will consider the equivalent abstract stochastic evolution equation

𝑑 𝑢 + 𝛼 2 + 𝐴 𝑢 𝜈 𝐴 𝑢 + 𝛼 2 𝐴 𝑢 + 𝐵 𝑢 , 𝑢 + 𝛼 2 𝐴 𝑢 𝑑 𝑡 = 𝐹 ( 𝑡 , 𝑢 ) 𝑑 𝑡 + 𝐺 ( 𝑡 , 𝑢 ) 𝑑 𝑊 , 𝑢 ( 0 ) = 𝑢 0 . ( 2 . 1 5 ) We now define the concept of strong solution of the problem (2.15) as follows.

Definition. By a strong solution of problem (2.15), we mean a stochastic process 𝑢 such that ( 1 ) 𝑢 ( 𝑡 ) is ℱ 𝑡 adapted for all 𝑡 ∈ [ 0 , 𝑇 ] ,( 2 ) 𝑢 ∈ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐷 ( 𝐴 3 / 2 ) ) ) ∩ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 , 𝐷 ( 𝐴 ) ) ) for all 1 ≤ 𝑝 < ∞ ,( 3 ) 𝑢 is weakly continuous with values in 𝐷 ( 𝐴 ) ,( 4 )P-a.s., the following integral equation holds: 𝑢 ( 𝑡 ) + 𝛼 2 𝐴 𝑢 ( 𝑡 ) , Φ + 𝜈 𝑡 0 𝑢 ( 𝑠 ) + 𝛼 2 𝐴 𝑢 ( 𝑠 ) , 𝐴 Φ 𝑑 𝑠 + 𝑡 0 𝐵 𝑢 ( 𝑠 ) , 𝑢 ( 𝑠 ) + 𝛼 2 = 𝑢 𝐴 𝑢 ( 𝑠 ) , Φ 𝑑 𝑠 0 + 𝛼 2 𝐴 𝑢 0 + , Φ 𝑡 0 ( 𝐹 ( 𝑠 , 𝑢 ( 𝑠 ) ) , Φ ) 𝑑 𝑠 + 𝑡 0 ( 𝐺 ( 𝑠 , 𝑢 ( 𝑠 ) ) , Φ ) 𝑑 𝑊 ( 𝑠 ) ( 2 . 1 6 ) for all Φ ∈ 𝒱 , and 𝑡 ∈ [ 0 , 𝑇 ] .

Notation 1. In this paper, weak convergence is denoted by ⇀ and strong convergence by → .

Our first result of this paper is the following.

Theorem 2.3 (existence and uniqueness). Suppose that the hypotheses (2.13) hold, and 𝑢 0 ∈ 𝐿 2 ( Ω , ℱ 0 , 𝑃 ; 𝐷 ( 𝐴 ) ) . Then problem (2.15) has a solution in the sense of Definition 2.2. The solution is unique almost surely and has in 𝐷 ( 𝐴 ) almost surely continuous trajectories.

We also prove that the sequence ( 𝑢 𝑛 ) of our Galerkin approximation (see (3.1) below) approximates the solution 𝑢 of the 3 𝐷 stochastic Leray- 𝛼 model in mean square.

This is the object of the second result of the paper.

Theorem 2.4 (Convergence results). Under the hypotheses of Theorem 2.3, the following convergences hold: 𝐸 𝑡 0 ‖ ‖ 𝑢 𝑛 ( ‖ ‖ 𝑠 ) − 𝑢 ( 𝑠 ) 2 𝐷 ( 𝐴 3 / 2 ) ‖ ‖ 𝑢 𝑑 𝑠 ⟶ 0 f o r 𝑛 ⟶ ∞ , 𝐸 𝑛 ( ‖ ‖ 𝑡 ) − 𝑢 ( 𝑡 ) 2 𝐷 ( 𝐴 ) ⟶ 0 , 𝑛 ⟶ ∞ ( 2 . 1 7 ) for all 𝑡 ∈ [ 0 , 𝑇 ] .

Remark. Theorems 2.3 and 2.4 are also true if one assumes measurable Lipschitz mappings 𝐹 ∶ Ω × ( 0 , 𝑇 ) × 𝐷 ( 𝐴 3 / 2 ) → 𝐻 and 𝐺 : Ω × ( 0 , 𝑇 ) × 𝐷 ( 𝐴 ) → 𝐻 ⊗ 𝑚 .

Remark. For the existence of the pressure, we can use a generalization of the Rham's theorem for processes (see [19, Theorem 4.1, Remark 4.3]). See also [6, page 15].

3. Galerkin Approximations and A Priori Estimates

We now introduce the Galerkin scheme associated to the original equation (2.15) and establish some uniform estimates.

3.1. The Approximate Equation

Let { 𝑤 𝑗 } ∞ 𝑗 = 1 be an orthonormal basis of 𝐻 consisting of eigenfunctions of the operator 𝐴 . Denote 𝐻 𝑛 = s p a n { 𝑤 1 , … , 𝑤 𝑛 } and let 𝑃 𝑛 be the 𝐿 2 -orthogonal projection from 𝐻 onto 𝐻 𝑛 .

We look for a sequence 𝑢 𝑛 ( 𝑡 ) in 𝐻 𝑛 solutions of the following initial value problem:

𝑑 𝑣 𝑛 + 𝜈 𝐴 𝑣 𝑛 + 𝑃 𝑛 𝐵 𝑢 𝑛 , 𝑣 𝑛 𝑑 𝑡 = 𝑃 𝑛 𝐹 𝑡 , 𝑢 𝑛 𝑑 𝑡 + 𝑃 𝑛 𝐺 𝑡 , 𝑢 𝑛 𝑢 𝑑 𝑊 , 𝑛 ( 0 ) = 𝑃 𝑛 𝑢 0 , 𝑣 𝑛 = 𝑢 𝑛 + 𝛼 2 𝐴 𝑢 𝑛 . ( 3 . 1 ) By the theory of stochastic differential equations (see [20–23]), there is a unique continuous ( ℱ 𝑡 ) -adapted process 𝑢 𝑛 ( 𝑡 ) ∈ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐻 𝑛 ) ) of (3.1).

We next establish some uniform estimates on 𝑢 𝑛 and 𝑣 𝑛 .

3.2. A Priori Estimates

Throughout this section 𝐶 , 𝐶 𝑖 ( 𝑖 = 1 , … ) denote positive constants independent of 𝑛 and 𝛼 .

Lemma 3.1. 𝑢 𝑛 and 𝑣 𝑛 satisfy the following a priori estimates: 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑣 𝑛 | | ( 𝑠 ) 2 + 4 𝜈 𝛽 𝐸 𝑇 0 ‖ ‖ 𝑣 𝑛 ‖ ‖ ( 𝑠 ) 2 𝑑 𝑠 ≤ 𝐶 1 , 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑢 𝑛 | | ( 𝑠 ) 2 ≤ 𝐶 2 , 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 ‖ ‖ 𝑢 𝑛 ‖ ‖ ( 𝑠 ) 2 < 𝐶 3 2 𝛼 2 , 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝐴 𝑢 𝑛 | | ( 𝑠 ) 2 ≤ 𝐶 4 𝛼 4 , 𝐸 𝑇 0 ‖ ‖ 𝑢 𝑛 ‖ ‖ ( 𝑠 ) 2 𝑑 𝑠 ≤ 𝐶 5 , 𝐸 𝑇 0 | | 𝐴 𝑢 𝑛 | | ( 𝑠 ) 2 𝐶 𝑑 𝑠 ≤ 6 2 𝛼 2 , 𝐸 𝑇 0 | | 𝐴 3 / 2 𝑢 𝑛 | | ( 𝑠 ) 2 𝐶 𝑑 𝑠 ≤ 7 𝛼 4 . ( 3 . 2 )

Proof. To prove Lemma 3.1, it suffices to establish the first inequality and use the fact that | | 𝑣 𝑛 | | 2 = | | 𝑢 𝑛 + 𝛼 2 𝐴 𝑢 𝑛 | | 2 = | | 𝑢 𝑛 | | 2 + 2 𝛼 2 ‖ ‖ 𝑢 𝑛 ‖ ‖ 2 + 𝛼 4 | | 𝐴 𝑢 𝑛 | | 2 , ‖ ‖ 𝑣 𝑛 ‖ ‖ 2 = ‖ ‖ 𝑢 𝑛 ‖ ‖ 2 + 2 𝛼 2 | | 𝐴 𝑢 𝑛 | | 2 + 𝛼 4 | | 𝐴 3 / 2 𝑢 𝑛 | | 2 . ( 3 . 3 ) By Ito’s formula, we have from (3.1) 𝑑 | | 𝑣 𝑛 | | ( 𝑡 ) 2 + 2 𝜈 ⟨ 𝐴 𝑣 𝑛 , 𝑣 𝑛 ⟩ 𝑉 ′ + 𝐵 𝑢 𝑛 , 𝑣 𝑛 , 𝑣 𝑛 𝑉 ′ = 𝑑 𝑡 2 𝐹 𝑡 , 𝑢 𝑛 , 𝑣 𝑛 + | | 𝑃 𝑛 𝐺 𝑡 , 𝑢 𝑛 | | 2 𝐺 𝑑 𝑡 + 2 𝑡 , 𝑢 𝑛 , 𝑣 𝑛 𝑑 𝑊 . ( 3 . 4 ) But then, taking into account (2.4), (2.2) and the fact that 𝐹 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 | | 𝑣 ( 𝑠 ) ≤ 𝐶 1 + 𝑛 | | ( 𝑠 ) 2 , | | 𝑃 𝑛 𝐺 𝑠 , 𝑢 𝑛 | | ( 𝑠 ) 2 | | 𝑣 ≤ 𝐶 1 + 𝑛 | | ( 𝑠 ) 2 , ( 3 . 5 ) we deduce from (3.4) that | | 𝑣 𝑛 ( | | 𝑡 ) 2 + 2 𝜈 𝛽 𝑡 0 ‖ ‖ 𝑣 𝑛 ‖ ‖ ( 𝑠 ) 2 ≤ | | 𝑣 𝑑 𝑠 𝑛 | | ( 0 ) 2 + 𝐶 2 𝑇 + 𝐶 3 𝑡 0 | | 𝑣 𝑛 | | ( 𝑠 ) 2 𝑑 𝑠 + 2 𝑡 0 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 ( 𝑠 ) 𝑑 𝑊 ( 𝑠 ) . ( 3 . 6 ) For each integer 𝑁 > 0 , consider the ℱ 𝑡 -stopping time 𝜏 𝑁 defined by 𝜏 𝑁 | | 𝑣 = i n f 𝑡 ∶ 𝑛 | | ( 𝑡 ) 2 ≥ 𝑁 2 ∧ 𝑇 . ( 3 . 7 ) It follows from (3.6) that s u p 𝑠 ∈ [ 0 , 𝑡 ∧ 𝜏 𝑁 ] | | 𝑣 𝑛 ( | | 𝑠 ) 2 + 2 𝜈 𝛽 𝑡 ∧ 𝜏 𝑁 0 ‖ ‖ 𝑣 𝑛 ‖ ‖ ( 𝑠 ) 2 | | 𝑣 𝑑 𝑠 ≤ 𝑛 ( | | 0 ) 2 + 𝐶 8 𝑇 + 𝐶 9 𝑡 ∧ 𝜏 𝑁 0 | | 𝑣 𝑛 ( | | 𝑠 ) 2 𝑑 𝑠 + 2 s u p 𝑠 ∈ [ 0 , 𝑡 ∧ 𝜏 𝑁 ] | | | | 𝑠 0 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 ( | | | | 𝑠 ) 𝑑 𝑊 ( 𝑠 ) ( 3 . 8 ) for all 𝑡 ∈ ( 0 , 𝑇 ) and all 𝑁 , 𝑛 ≥ 1 . Taking expectation in (3.8), by Doob’s inequality it holds 𝐸 s u p 𝑠 ∈ [ 0 , 𝑡 ∧ 𝜏 𝑁 ] 𝑠 0 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 ( 𝑠 ) 𝑑 𝑊 ( 𝑠 ) ≤ 3 𝐸 𝑡 ∧ 𝜏 𝑁 0 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 ( 𝑠 ) 2 𝑑 𝑠 1 / 2 ≤ 3 𝐸 𝑡 ∧ 𝜏 𝑁 0 | | 𝐺 𝑠 , 𝑢 𝑛 | | ( 𝑠 ) 2 | | 𝑣 𝑛 | | ( 𝑠 ) 2 𝑑 𝑠 1 / 2 ≤ 1 2 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑡 ∧ 𝜏 𝑁 | | 𝑣 𝑛 | | ( 𝑠 ) 2 + 𝐶 1 0 𝑇 + 𝐶 1 1 𝐸 𝑡 ∧ 𝜏 𝑁 0 | | 𝑣 𝑛 | | ( 𝑠 ) 2 𝑑 𝑠 . ( 3 . 9 ) Next using Gronwall’s lemma, it follows that there exists a constant 𝐶 1 depending on 𝑇 , 𝐶 such that, for all 𝑛 ≥ 1 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑣 𝑛 | | ( 𝑠 ) 2 + 4 𝜈 𝛽 𝐸 𝑇 0 ‖ ‖ 𝑣 𝑛 ‖ ‖ ( 𝑠 ) 2 𝑑 𝑠 ≤ 𝐶 1 . ( 3 . 1 0 )

The following result is related to the higher integrability of 𝑢 𝑛 and 𝑣 𝑛 .

Lemma 3.2. One has 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑣 𝑛 | | ( 𝑠 ) 𝑝 ≤ 𝐶 𝑝 , 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑢 𝑛 | | ( 𝑠 ) 𝑝 ≤ 𝐶 𝑝 , ( 3 . 1 1 ) 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 ‖ ‖ 𝑢 𝑛 ‖ ‖ ( 𝑠 ) 𝑝 ≤ 𝐶 𝑝 𝛼 𝑝 , ( 3 . 1 2 ) 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑢 𝑛 | | ( 𝑠 ) 𝑝 𝐷 ( 𝐴 ) ≤ 𝐶 𝑝 𝛼 2 𝑝 ( 3 . 1 3 ) for all 1 ≤ 𝑝 < ∞ .

Proof. By Ito's formula, we have for 4 ≤ 𝑝 < ∞ 𝑑 | | 𝑣 𝑛 | | ( 𝑡 ) 𝑝 / 2 = 𝑝 2 | | 𝑣 𝑛 | | ( 𝑡 ) 𝑝 / 2 − 2 × − 𝜈 ⟨ 𝐴 𝑣 𝑛 , 𝑣 𝑛 ⟩ 𝑉 ′ − 𝐵 𝑢 𝑛 , 𝑣 𝑛 , 𝑣 𝑛 𝑉 ′ + 𝐹 𝑡 , 𝑢 𝑛 , 𝑣 𝑛 + 𝑝 − 4 4 𝐺 𝑡 , 𝑢 𝑛 , 𝑣 𝑛 2 | | 𝑣 𝑛 | | ( 𝑡 ) 2 + 𝑝 𝑑 𝑡 2 | | 𝑣 𝑛 | | ( 𝑡 ) 𝑝 / 2 − 2 𝐺 𝑡 , 𝑢 𝑛 , 𝑣 𝑛 𝑑 𝑊 . ( 3 . 1 4 ) Taking into account (2.4) and the fact that | | 𝑣 𝑛 | | ( 𝑠 ) ( 𝑝 / 2 ) − 2 𝐹 𝑡 , 𝑢 𝑛 , 𝑣 𝑛 | | 𝑣 ≤ 𝐶 1 + 𝑛 | | ( 𝑠 ) 𝑝 / 2 𝐺 ( Y o u n g ' s i n e q u a l i t y ) , 𝑠 , 𝑢 𝑛 , 𝑣 𝑛 2 | | 𝑣 𝑛 | | ( 𝑠 ) 2 | | 𝑣 ≤ 𝐶 1 + 𝑛 | | ( 𝑠 ) 2 , ( 3 . 1 5 ) we deduce from (3.14) that | | 𝑣 𝑛 ( | | 𝑡 ) 𝑝 / 2 ≤ | | 𝑣 𝑛 ( | | 0 ) 𝑝 / 2 + 𝐶 𝑡 0 | | 𝑣 1 + 𝑛 ( | | 𝑠 ) 𝑝 / 2 𝑝 𝑑 𝑠 + 2 𝑡 0 | | 𝑣 𝑛 ( | | 𝑠 ) 𝑝 / 2 − 2 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 ( 𝑠 ) 𝑑 𝑊 ( 𝑠 ) . ( 3 . 1 6 ) Taking the supremum, the square, and the mathematical expectation in (3.16), and owing to the Martingale's inequality it holds 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | | | 𝑠 0 | | 𝑣 𝑛 | | ( 𝑠 ) 𝑝 / 2 − 2 ( 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 | | | | ( 𝑠 ) ) 𝑑 𝑊 ( 𝑠 ) 2 ≤ 4 𝐸 𝑇 0 | | 𝑣 𝑛 | | ( 𝑠 ) 𝑝 − 4 𝐺 𝑠 , 𝑢 𝑛 ( 𝑠 ) , 𝑣 𝑛 ( 𝑠 ) 2 𝑑 𝑠 ≤ 4 𝐶 𝐸 𝑇 0 | | 𝑣 1 + 𝑛 | | ( 𝑠 ) 𝑝 𝑑 𝑠 . ( 3 . 1 7 ) Applying Gronwall’s lemma, it follows that there exists a constant 𝐶 𝑝 , such that 𝐸 s u p 0 ≤ 𝑠 ≤ 𝑇 | | 𝑣 𝑛 | | ( 𝑠 ) 𝑝 ≤ 𝐶 𝑝 ( 3 . 1 8 ) for all 𝑝 ≥ 4 . With this being proved for any 𝑝 ≥ 4 , it is subsequently true for any 1 ≤ 𝑝 < ∞ .
Other inequalities are deduced from the relation
| | 𝑣 𝑛 | | ( 𝑠 ) 2 = | | 𝑢 𝑛 | | ( 𝑠 ) 2 + 2 𝛼 2 ‖ ‖ 𝑢 𝑛 ‖ ‖ ( 𝑠 ) 2 + 𝛼 4 | | 𝐴 𝑢 𝑛 | | ( 𝑠 ) 2 . ( 3 . 1 9 )

We also have the following.

Lemma. One has 𝐸 𝑇 0 ‖ ‖ 𝑣 𝑛 ‖ ‖ ( 𝑠 ) 2 𝑑 𝑠 𝑝 ≤ 𝐶 𝑝 f o r 1 ≤ 𝑝 < ∞ . ( 3 . 2 0 )

Proof. The proof is derived from (4.46), Martingale’s inequality, and Lemma 3.2.

4. Proof of Theorem 2.3

4.1. Existence

With the uniform estimates on the solution of the Galerkin approximations in hand, we proceed to identify a limit 𝑢 . This stochastic process is shown to satisfy a stochastic partial differential equations (see (4.2)) with unknown terms corresponding to the nonlinear portions of the equation. Next, using the properties of stopping times and some basic convergence principles from functional analysis, we identify the unknown portions.

We will split the proof of the existence into two steps.

4.1.1. Taking Limits in the Finite-Dimensional Equations

Lemma (limit system). Under the hypotheses of Theorem 2.3, there exist adapted processes 𝑢 , 𝐵 ∗ , 𝐹 ∗ , and 𝐺 ∗ with the regularity, 𝑢 ∈ 𝐿 𝑝 Ω , ℱ , 𝑃 ; 𝐿 2 𝐴 0 , 𝑇 ; 𝐷 3 / 2 ∩ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝐷 ( 𝐴 ) ) ) , 𝑣 ∈ 𝐿 𝑝 Ω , ℱ , 𝑃 ; 𝐿 2 , 𝐵 ( 0 , 𝑇 ; 𝑉 ) 𝑣 ∈ 𝐶 ( 0 , 𝑇 ; 𝐻 ) 𝑎 . 𝑠 . , 𝑢 ∈ 𝐶 ( 0 , 𝑇 ; 𝐷 ( 𝐴 ) ) 𝑎 . 𝑠 . , ∗ ∈ 𝐿 2 Ω , ℱ , 𝑃 ; 𝐿 2 0 , 𝑇 ∶ 𝑉 , 𝐹 ∗ ∈ 𝐿 2 Ω , ℱ , 𝑃 ; 𝐿 2 , 𝐺 ( 0 , 𝑇 ; 𝐻 ) ∗ ∈ 𝐿 2 Ω , ℱ , 𝑃 ; 𝐿 2 0 , 𝑇 ; 𝐻 ⊗ 𝑚 , ( 4 . 1 ) such that 𝑢 , 𝐵 ∗ , 𝐹 ∗ , and 𝐺 ∗ satisfy 𝑣 ( 𝑡 ) + 𝜈 𝑡 0 𝐴 𝑣 ( 𝑠 ) 𝑑 𝑠 + 𝑡 0 𝐵 ∗ ( 𝑠 ) 𝑑 𝑠 = 𝑣 ( 0 ) + 𝑡 0 𝐹 ∗ ( 𝑠 ) 𝑑 𝑠 + 𝑡 0 𝐺 ∗ ( 𝑠 ) 𝑑 𝑊 ( 𝑠 ) ( 4 . 2 ) where 𝑣 ( 𝑡 ) = 𝑢 ( 𝑡 ) + 𝛼 2 𝐴 𝑢 ( 𝑡 ) and 1 ≤ 𝑝 < ∞ .

Remark. We use the following elementary facts regarding weakly convergent sequences in the proof below. (i)Let 𝑆 1 and 𝑆 2 be Banach spaces and let 𝐿 ∶ 𝑆 1 → 𝑆 2 be a continuous linear operator. If ( 𝑥 𝑛 ) is a sequence in 𝑆 1 such that 𝑥 𝑛 ⇀ 𝑥 (where 𝑥 ∈ 𝑆 1 ), then 𝐿 ( 𝑥 𝑛 ) ⇀ 𝐿 ( 𝑥 ) .(ii)If 𝑆 is Banach space and if ( 𝑥 𝑛 ) is a sequence from 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑆 ) ) , which converges weakly to 𝑥 in 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑆 ) ) , then for 𝑛 → ∞ the following assertions are true: 𝑡 0 𝑥 𝑛 ( 𝑠 ) 𝑑 𝑠 ⇀ 𝑡 0 𝑥 ( 𝑠 ) 𝑑 𝑠 , 𝑡 0 𝑥 𝑛 ( 𝑠 ) 𝑑 𝑊 ( 𝑠 ) ⇀ 𝑡 0 𝑥 ( 𝑠 ) 𝑑 𝑊 ( 𝑠 ) ( 4 . 3 ) in 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑆 ) ) .

Proof. Using (2.8) and Hölder’s inequality, we have 𝐸 𝑇 0 ‖ ‖ 𝑃 𝑛 𝐵 ( 𝑢 𝑛 ( 𝑡 ) , 𝑣 𝑛 ‖ ‖ ( 𝑡 ) 2 𝑉 ′ ≤ 𝐶 𝐸 s u p 𝑡 ∈ [ 0 , 𝑇 ] ‖ ‖ 𝑢 𝑛 ‖ ‖ ( 𝑡 ) 4 1 / 2 𝐸 𝑇 0 ‖ ‖ 𝑣 𝑛 ‖ ‖ ( 𝑡 ) 2 𝑑 𝑡 2 1 / 2 . ( 4 . 4 ) The later quantity is uniformly bounded as a consequence of Lemmas 3.2, 3.3. From (4.4), we can deduce that the sequence 𝑃 𝑛 𝐵 ( 𝑢 𝑛 , 𝑣 𝑛 ) is bounded in 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑉 ) ) . On the other hand, from Lemmas 3.1, 3.2, 3.3 and the Lipschitz conditions on 𝐹 and 𝐺 , we have that the sequence 𝑢 𝑛 is bounded in 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐷 ( 𝐴 3 / 2 ) ) ∩ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝐷 ( 𝐴 ) ) , the sequence 𝑣 𝑛 is bounded in 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑉 ) ) ∩ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝐻 ) ) , the sequence 𝑣 𝑛 ( 0 ) is bounded in 𝐿 2 ( Ω , ℱ 0 , 𝑃 ; 𝐻 ) , the sequence 𝑢 𝑛 ( 0 ) is bounded in 𝐿 2 ( Ω , ℱ 0 , 𝑃 ; 𝐷 ( 𝐴 ) ) , the sequence 𝑃 𝑛 𝐹 ( 𝑡 , 𝑢 𝑛 ) is bounded in 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐻 ) ) , and 𝑃 𝑛 𝐺 ( 𝑡 , 𝑢 𝑛 ) is bounded in 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐻 ⊗ 𝑚 ) ) .
Thus with Alaoglu's theorem, we can ensure that there exists a subsequence { 𝑢 𝑛 ′ } ⊂ { 𝑢 𝑛 } , and seven elements 𝑢 ∈ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐷 ( 𝐴 3 / 2 ) ) ) ∩ 𝐿 𝑝 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝐷 ( 𝐴 ) ) ) , 𝑣 ∈ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑉 ) ) ∩ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 ∞ ( 0 , 𝑇 ; 𝐻 ) ) , 𝐵 ∗ ∈ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝑉 ) ) , 𝐹 ∗ ∈ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐻 ) ) , 𝜌 1 ∈ 𝐿 2 ( Ω , ℱ 0 , 𝐻 ) , 𝜌 2 ∈ 𝐿 2 ( Ω , ℱ 0 , 𝐷 ( 𝐴 ) ) and 𝐺 ∗ ∈ 𝐿 2 ( Ω , ℱ , 𝑃 ; 𝐿 2 ( 0 , 𝑇 ; 𝐻 ⊗ 𝑚 ) ) such that:
𝑢 𝑛 ′ ⇀ 𝑢 i n 𝐿 𝑝 Ω , ℱ , 𝑃 ; 𝐿 2