# On Stochastic Models Describing the Motions of Randomly Forced Linear Viscoelastic Fluids

- PA Razafimandimby
^{1}Email author

**2010**:932053

https://doi.org/10.1155/2010/932053

© P. A. Razafimandimby. 2010

**Received: **19 August 2009

**Accepted: **16 March 2010

**Published: **26 April 2010

## Abstract

This paper is devoted to the analysis of stochastic equations describing the motions of a large class of incompressible linear viscoelastic fluids in two-dimensional subject to periodic boundary condition and driven by random external forces. To do so we distinguish two cases, and for each case a global existence result of probabilistic weak solution is expounded in this paper. We also prove that under suitable hypotheses on the external random forces the solution turns out to be unique. As concrete examples, we consider the stochastic equations for the Maxwell and Oldroyd fluids that are of great importance in the investigation towards the understanding of the elastic turbulence.

## Keywords

## 1. Introduction

The study of turbulent flows has attracted many prominent researchers from different fields of contemporary sciences for ages. For in-depth coverage of the deep and fascinating investigations undertaken in this field, the abundant wealth of results obtained, and remarkable advances achieved we refer to the monographs [1–3] and references therein. Recent study, see, for instance [4], has showed that the non-Newtonian elastic turbulence can be well understood on basis of known viscoelastic models such as the Oldroyd fluids or the Maxwell fluids. Indeed, by computational investigations of the two-dimensional periodic Oldroyd-B model the authors in [4] found that there is a considerable agreement between their numerical results and the experimental observations of elastic turbulence.

The hypothesis relating the turbulence to the "randomness of the background field" is one of the motivations of the study of stochastic version of equations governing the motion of fluids flows. The introduction of random external forces of noise type reflects (small) irregularities that give birth to a new random phenomenon, making the problem more realistic. Such approach in the mathematical investigation for the understanding of the Newtonian turbulence phenomenon was pioneered by Bensoussan and Temam in [5] where they studied the Stochastic Navier-Stokes Equation (SNSE). Since then stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations which have generated several important results. We refer to [6–13], just to cite a few. Similar investigations for Non-Newtonian elastic fluids have not almost been undertaken except in very few works; we refer, for instance, to [14–19] for some example of computational studies of stochastic models of polymeric fluids and to [20–23] for their mathematical analysis. It should be noted that the models investigated in these papers occur very naturally from the kinetic theory of polymer dynamics. Indeed they arise from the reformulation of Fokker-Planck or diffusion equations as stochastic differential equations (3.45). We also notice that they model some nonlinear viscoelastic models such as the FENE models which are very different from the models we shall treat in this paper. We refer to volume 2 of [24] for the conventional approach to kinetic theory which consists of deriving the deterministic partial differential equations for the polymer configurational distribution function (diffusion equation) and to volume 1 of [24] for the existing linear and nonlinear viscoelastic models.

and the operator is a continuous mapping satisfying some hypotheses (see (2.22)–(2.24)). The problem (1.1) also can be taken as a turbulent version of linear viscoelastic models for polymeric fluids. For some examples of classical models of turbulence, we refer to [2, 4, 19, 25] and references therein.

The mathematical works on some linear viscoelastic fluids undertaken by the Soviet mathematician Oskolkov in [26–28] and by Ladyzhenskaya in [29] have influenced the emergence of the paper [30] where a global solvability result of the deterministic counterpart of the system (1.1), (1.2) (resp., (1.1), (1.3)) subject to the periodic boundary condition (resp., nonslip boundary condition) was given. To the best of our knowledge similar investigations for the two general stochastic models (1.1), (1.2) and (1.1), (1.3) have not been undertaken yet. The purpose of this paper is to prove that under suitable conditions on , and each of our stochastic model is well posed (see Theorems 3.3, 3.4, 4.2, and 4.3). In view of the technical difficulties involved, we provide full details of the proof of our results. Due to nontrivial difficulties that arise from the nature of the nonlinearities involved in (1.1) other mathematical issues such as existence, uniqueness of the invariant measure, and its ergodicity are beyond of the scope of this work; we leave these questions for future investigation.

The layout of this paper is as follows. In addition to the current introduction this article consists of three other sections. In Section 2 we give some notations, necessary backgrounds of probabilistic or analytic nature. Section 3 is devoted to the detailed analysis of the problem (1.1), (1.2). We prove the existence and pathwise uniqueness of its probabilistic weak solution which yields the existence of a unique probabilistic strong solution. In the very same section we consider the stochastic equations for randomly forced generalized Maxwell fluids as a concrete example. In Section 4 we only state the main theorems related to (1.1), (1.3) and apply the obtained results to the stochastic model for the generalized viscoelastic Oldroyd fluids; we refer to the previous section for the details of the proofs.

## 2. Preliminaries and Notations

We refer to [32] (see also [1, 33]) for more details about these spaces. We proceed with the definitions of additional spaces frequently used in this paper.

where denotes the space of infinitely differentiable periodic function with period .

for any and . Here denotes the duality product .

Next we define some probabilistic evolution spaces necessary throughout the paper. Let be a given stochastic basis; that is, is a complete probability space and is an increasing sub- -algebras of such that contains every -null subset of . For any real Banach space , for any we denote by the space of processes with values in defined on such that

(1) is measurable with respect to and for each , is -measurable;

where denotes the mathematical expectation with respect to the probability measure .

We refer to [34, 35] (see also [36]) for further reading on probability theory and stochastic calculus.

The family is said to be tight if, for any , there exists a compact set such that , for every .

We have the well-known result.

Theorem 2.1 (Prokhorov).

Assume that is a Polish space; then the family is relatively compact if and only if it is tight.

We will use the following useful theorem due to Skorokhod.

Theorem 2.2 (Skorokhod).

For any sequence of probability measures on which converges to a probability measure , there exists a probability space and random variables , with values in such that the probability law of (resp., ) is (resp., ) and -a.s.

We refer to [36] for the proofs of these two theorems.

The following result is very important in Section 3.2.2 where we prove a probabilistic compactness result; its proof can be found in [37].

Theorem 2.3.

for any is compact in for any .

We assume that is a symmetric tensor-valued continuous mapping which satisfies the following.

Remark 2.4.

We also notice that (2.24) and (2.23) are equivalent if is linear.

## 3. Analysis of the Stochastic Equation of the Type (1.1), (1.2)

In this section we investigate the stochastic equations (1.1), (1.2). The first subsection is devoted to the statement of the main results which is going to be proved in the second subsection.

### 3.1. Hypotheses and Statement of the Main Results

Throughout this section we suppose the following.

for almost everywhere and for any .

(HYP 4)In addition to (2.22)–(2.24) we assume furthermore that

Remark 3.1.

where is an arbitrary constant vector, and is a vector field.

Karazeeva remarked in [30, Section ] that when and , , commute, then (3.4) is a consequence of (2.23). The condition (3.4) is met when is given by the equation in Remark 2.4.

We introduce the concept of the solution of the problem (1.1), (1.2).

Definition 3.2.

where

(1) is a complete probability space, and is a filtration on ;

(2) is an -dimensional -standard Wiener process;

(4)for almost all , is -measurable;

We have the following.

Theorem 3.3.

If and if the hypotheses (HYP 1), (HYP 2), and (HYP 4) hold, then the problem (1.1), (1.2) has a solution in the sense of the above definition. Moreover, almost surely the paths of the solution are - (resp., -), valued weakly (resp., strongly) continuous.

Theorem 3.4.

Assume that (HYP 1)–(HYP 4) hold and let and be two probabilistic weak solutions of (1.1), (1.2) starting with the same initial condition and defined on the same stochastic basis with the same Winer process . If one sets , then one has almost surely.

### 3.2. Proof of Theorems 3.3 and 3.4

This subsection is devoted to the proof of the existence and uniqueness results stated in the previous subsection. We split the proof into four subsections. The proof of the existence theorem is inspired by the works [6, 30] (see also [10]). Throughout this subsection will designate a positive constant which depends only on the data ( ).

#### 3.2.1. The Approximate Solution and Some A Priori Estimates

In this subsection we derive crucial a priori estimates from the Galerkin approximation. They will serve as a toolkit for the proof of the Theorem 3.3.

which is a system of stochastic ordinary differential equations with continuous coefficients. From the existence theorem stated in [39, page 59] (see also [35, Theorem page 323]) we infer the existence of continuous functions on . Global existence will follow from a priori estimates for .

Lemma 3.5.

Proof.

We infer from this, (3.27), and the monotonicity of that a.s. as was required. Since the constant in (3.27) is independent of and , Fatou's theorem completes the proof of the lemma.

The estimate of Lemma 3.5 is not sufficient to pass to the limit in the nonlinear term. We still need to derive some additional crucial but nontrivial inequalities.

Lemma 3.6.

Proof.

Owing to (2.6) the proof of the lemma is finished.

The following result is central in the proof of the forthcoming tightness property of the Galerkin solution.

Lemma 3.7.

Proof.

To complete the proof we use the same argument for negative values of .

#### 3.2.2. Tightness Property and Application of Prokhorov's and Skorohod's Theorems

for any sequences such that as and . The following result is a particular case of Theorem 2.3 (see also [40, Proposition , page 45] for a similar result).

Lemma 3.8.

Lemma 3.9.

Proof.

A convenient choice of and completes the proof of the claim, and hence the proof of the lemma.

Moreover, the probability law of is and that of is .

For the filtration , it is enough to choose .

By the same argument as in [40, Section ] (see also [41, Section ]) we can prove that the limit process is a standard -dimensional Wiener process defined on .

Theorem 3.10.

for almost all , for any and .

Proof.

The proof follows the same lines as in [6, Section ] (see also [41, Section ]), and so we omit it.

#### 3.2.3. Passage to the Limit

Combining all these results and passing to the limit in (3.60), we see that satisfies (3.8) which holds for almost all , for all . This proves the first part of Theorem 3.3. By arguing as in [43] (Chapter 2, Lemma ) we get the continuity result stated in Theorem 3.3.

#### 3.2.4. Proof of the Uniqueness of the Solution

for any . Since , then this completes the proof of Theorem 3.4.

### 3.3. Example: The Stochastic Equation for the Maxwell Fluids

is assumed to be positive. Here the numbers designate the roots of the polynomial . The result in [30] shows that satisfies (2.22)–(2.24) and (3.4). Hence the results in Theorems 3.3 and 3.4 can be applied to the stochastic equations (1.1)-(1.2) and (3.81) for the Maxwell fluids provided that the assumptions (HYP 1)–(HYP 4) hold.

## 4. Stochastic Equation of Type (1.1), (1.3)

This section is devoted to the investigation of (1.1), (1.3). We omit the details of the proofs since they can be derived from similar ideas used in Section 3. We state our main results in the first subsection and we give a concrete example in the second subsection. For this section we suppose following.

### 4.1. Statement of the Main Results

We introduce the concept of the solution of the problem (1.1), (1.3).

Definition 4.1.

where

(1) is a complete probability space, is a filtration on ;

(2) is an -dimensional -standard Wiener process;

(4)for almost all , is -measurable;

We have the following.

Theorem 4.2.

If and if the hypotheses (AF)-(AG) hold, then the problem (1.1), (1.3) has a solution in the sense of the above definition. Moreover is strongly (resp., weakly) continuous in (resp., ) with probability one.

Proof.

The proof follows from the Galerkin method; and the compactness method, the procedure is very similar to the proof of Theorem 3.3, and it is even easier. We just formally derive the crucial estimates.

We also have the uniqueness result whose proof follows from similar arguments used in Theorem 3.4.

Theorem 4.3.

Assume that (AF)–(ASFG) hold and let and be two probabilistic weak solutions of (1.1), (1.3) starting with the same initial condition and defined on the same stochastic basis . If one sets , then one has almost surely.

### 4.2. Application to the Oldroyd Fluids

and that satisfies the assumption (2.22)–(2.24). Therefore Theorems 4.2 and 4.3 hold for the Oldroyd fluid provided that the assumptions on and (see (AF)–(ASFG)) are valid.

Remark 4.4.

## Declarations

### Acknowledgments

The research of the author is supported by the University of Pretoria and the National Research Foundation South Africa. The author is very grateful to the referees for their insightful comments. He also thanks professor M. Sango for his invaluable comments and M. Ramaroson for her encouragement.

## Authors’ Affiliations

## References

- Foias C, Manley O, Rosa R, Temam R:
*Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications*.*Volume 83*. Cambridge University Press, Cambridge, UK; 2001:xiv+347.View ArticleMATHGoogle Scholar - McComb WD:
*The Physics of Fluid Turbulence, Oxford Engineering Science Series*.*Volume 25*. The Clarendon Press, Oxford University Press, New York, NY, USA; 1991:xxiv+572.MATHGoogle Scholar - Monin AS, Yaglom AM:
*Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. II*. Dover Publications, Mineola, NY, USA; 2007:xii+874.MATHGoogle Scholar - Berti S, Bistagnino A, Boffeta G, Celani A, Musacchio S: Two-dimensional elastic turbulence.
*Physical Review E*2008, 77: 1–4.View ArticleGoogle Scholar - Bensoussan A, Temam R: Equations stochastiques du type Navier-Stokes.
*Journal of Functional Analysis*1973, 13: 195–222. 10.1016/0022-1236(73)90045-1MathSciNetView ArticleMATHGoogle Scholar - Bensoussan A: Stochastic Navier-Stokes equations.
*Acta Applicandae Mathematicae*1995, 38(3):267–304. 10.1007/BF00996149MathSciNetView ArticleMATHGoogle Scholar - Capiński M, Cutland NJ: Navier-Stokes equations with multiplicative noise.
*Nonlinearity*1993, 6(1):71–78. 10.1088/0951-7715/6/1/005MathSciNetView ArticleMATHGoogle Scholar - Capiński M, Cutland NJ: Statistical solutions of stochastic Navier-Stokes equations.
*Indiana University Mathematics Journal*1994, 43(3):927–940. 10.1512/iumj.1994.43.43040MathSciNetView ArticleMATHGoogle Scholar - Da Prato G, Debussche A: 2D stochastic Navier-Stokes equations with a time-periodic forcing term.
*Journal of Dynamics and Differential Equations*2008, 20(2):301–335. 10.1007/s10884-007-9074-1MathSciNetView ArticleMATHGoogle Scholar - Flandoli F, Gatarek D: Martingale and stationary solutions for stochastic Navier-Stokes equations.
*Probability Theory and Related Fields*1995, 102(3):367–391. 10.1007/BF01192467MathSciNetView ArticleMATHGoogle Scholar - Fu X: Existence and stability of solutions for nonautonomous stochastic functional evolution equations.
*Journal of Inequalities and Applications*2009, 2009:-27.MathSciNetMATHGoogle Scholar - Mikulevicius R, Rozovskii BL: Stochastic Navier-Stokes equations for turbulent flows.
*SIAM Journal on Mathematical Analysis*2004, 35(5):1250–1310. 10.1137/S0036141002409167MathSciNetView ArticleMATHGoogle Scholar - Viot M:
*Sur les solutions faibles des Equations aux dérivées partielles stochastiques non linéaires, Thèse de Doctorat d'état*. Université Paris-Sud; 1975.Google Scholar - Hulsen MA, van Heel APG, van den Brule BHAA: Simulation of viscoelastic ows using Brownian configuration fields.
*Journal of Non-Newtonian Fluid Mechanics*1997, 70: 79–101. 10.1016/S0377-0257(96)01503-0View ArticleGoogle Scholar - Jourdain B, Lelièvre T, Le Bris C: Numerical analysis of micro-macro simulations of polymeric fluid flows: a simple case.
*Mathematical Models and Methods in Applied Sciences*2002, 12(9):1205–1243. 10.1142/S0218202502002100MathSciNetView ArticleMATHGoogle Scholar - Jourdain B, Le Bris C, Lelievre T: On a variance reduction technique for micromacro simulations of polymeric fluids.
*Journal of Non-Newtonian Fluid Mechanics*2004, 122: 91–106. 10.1016/j.jnnfm.2003.09.006View ArticleMATHGoogle Scholar - Li T, Vanden-Eijnden E, Zhang P, Weinan E: Stochastic models for polymeric fluids at small Deborah number.
*Journal of Non-Newtonian Fluid Mechanics*2004, 121: 117–125. 10.1016/j.jnnfm.2004.05.003View ArticleMATHGoogle Scholar - Öttinger HC:
*Stochastic Processes in Polymeric Fluids*. Springer, Berlin, Germany; 1996:xxiv+362.View ArticleMATHGoogle Scholar - Vincenzi D, Jin S, Bodenschatz E, Collins LR: Stretching of polymers in isotropic turbulence: a statistical closure.
*Physical Review Letters*2007, 98:-4.View ArticleGoogle Scholar - Bonito A, Clément P, Picasso M: Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows.
*Journal of Evolution Equations*2006, 6(3):381–398. 10.1007/s00028-006-0251-1MathSciNetView ArticleMATHGoogle Scholar - Jourdain B, Lelièvre T: Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. In
*Probabilistic Methods in Fluids*. Edited by: Davies IM, Jacob N, Truman A, Hassan O, Morgan K, Weatherill NP. World Scientific, River Edge, NJ, USA; 2003:205–223.View ArticleGoogle Scholar - Jourdain B, Lelièvre T, Le Bris C: Existence of solution for a micro-macro model of polymeric fluid: the FENE model.
*Journal of Functional Analysis*2004, 209(1):162–193. 10.1016/S0022-1236(03)00183-6MathSciNetView ArticleMATHGoogle Scholar - Jourdain B, Le Bris C, Lelièvre T, Otto F: Long-time asymptotics of a multiscale model for polymeric fluid flows.
*Archive for Rational Mechanics and Analysis*2006, 181(1):97–148. 10.1007/s00205-005-0411-4MathSciNetView ArticleMATHGoogle Scholar - Bird RB, Armstrong RC, Hassanger O:
*Dynamics of Polymeric Fluids, Vol. 1 and Vol. 2*. John Wiley & Sons, New York, NY, USA; 1987.Google Scholar - Becker T, Eckhardt B: Turbulence in a Maxwell fluid.
*Zeitschrift fur Physik B*1996, 101(3):461–468. 10.1007/s002570050234View ArticleGoogle Scholar - Oskolkov AP: Certain model nonstationary systems in the theory of non-Newtonian fluids.
*Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova*1975, 127: 32–57.MathSciNetGoogle Scholar - Oskolkov AP: On the theory of Maxwell fluids.
*Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI)*1981, 101: 119–127.MathSciNetMATHGoogle Scholar - Oskolkov AP: Initial-boundary value problems for equations of motion of Kelvin-Voight fluids and Oldroyd fluids.
*Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova*1988, 179: 126–164.MathSciNetGoogle Scholar - Ladyzhenskaya O: On global existence of weak solutions to some 2-dimensional initial-boundary value problems for Maxwell fluids.
*Applicable Analysis*1997, 65(3–4):251–255. 10.1080/00036819708840561MathSciNetView ArticleMATHGoogle Scholar - Karazeeva NA: Solvability of initial boundary value problems for equations describing motions of linear viscoelastic fluids. Journal of Applied Mathematics 2005, (1):59–80.Google Scholar
- Adams RA:
*Sobolev Spaces*. Academic Press, New York, NY, USA; 1975:xviii+268.Google Scholar - Temam R:
*Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics*.*Volume 66*. 2nd edition. SIAM, Philadelphia, Pa, USA; 1995:xiv+141.View ArticleGoogle Scholar - Constantin P, Foias C:
*Navier-Stokes Equations, Chicago Lectures in Mathematics*. University of Chicago Press, Chicago, Ill, USA; 1988:x+190.Google Scholar - Gikhman II, Skorohod AV:
*Stochastic Differential Equations*. Springer, Berlin, Germany; 1972.View ArticleGoogle Scholar - Karatzas I, Shreve SE:
*Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics*.*Volume 113*. Springer, New York, NY, USA; 1988:xxiv+470.View ArticleMATHGoogle Scholar - Da Prato G, Zabczyk J:
*Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications*.*Volume 44*. Cambridge University Press, Cambridge, UK; 1992:xviii+454.View ArticleMATHGoogle Scholar - Simon J: Compact sets in the space .
*Annali di Matematica Pura ed Applicata. Serie Quarta*1987, 146: 65–96.MathSciNetView ArticleMATHGoogle Scholar - Astarita G, Marucci G:
*Principles of Non-Newtonian Fluid Mechanics*. McGraw-Hill, London, UK; 1974.Google Scholar - Skorokhod AV:
*Studies in the Theory of Random Processes*. Addison-Wesley, Reading, Mass, USA; 1965:viii+199.Google Scholar - Bensoussan A: Some existence results for stochastic partial differential equations. In
*Stochastic Partial Differential Equations and Applications (Trento, 1990), Pitman Res. Notes Math. Ser.*.*Volume 268*. Longman Sci. Tech., Harlow, UK; 1992:37–53.Google Scholar - Razafimandimby PA, Sango M: Weak solutions of a stochastic model for two-dimensional second grade fluids.
*Boundary Value Problems*2010, 2010:-47.MathSciNetView ArticleMATHGoogle Scholar - Lions JL:
*Quelques Méthodes de Résolution des Problèmes aus Limites non Linéaires*. Etudes Mathématiques. Dunod; 1969.MATHGoogle Scholar - Pardoux E:
*Equations aux Dérivées Partielles Stochastiques Monotones, Thèse de Doctorat*. Université Paris-Sud; 1975.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.