- Research Article
- Open Access
On Stochastic Models Describing the Motions of Randomly Forced Linear Viscoelastic Fluids
© P. A. Razafimandimby. 2010
- Received: 19 August 2009
- Accepted: 16 March 2010
- Published: 26 April 2010
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.
- Stochastic Equation
- Stochastic Partial Differential Equation
- Standard Wiener Process
- Maxwell Fluid
- Martingale Inequality
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 , 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  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  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  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  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  have influenced the emergence of the paper  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.
If is the norm on , then
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 .
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  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 .
for any is compact in for any .
We assume that is a symmetric tensor-valued continuous mapping which satisfies the following.
We also notice that (2.24) and (2.23) are equivalent if is linear.
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 .
for any .
(HYP 4)In addition to (2.22)–(2.24) we assume furthermore that
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).
(1) is a complete probability space, and is a filtration on ;
(2) is an -dimensional -standard Wiener process;
(4)for almost all , is -measurable;
for any and .
We have the following.
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.
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 ). 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 .
for any and .
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.
for any and .
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.
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).
The set is compact in .
That is, for any , .
The family is tight in .
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 .
for almost all , for any and .
3.2.3. Passage to the Limit
for any and as .
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  (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  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.
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.
for all and for any .
for any .
4.1. Statement of the Main Results
We introduce the concept of the solution of the problem (1.1), (1.3).
(1) is a complete probability space, is a filtration on ;
(2) is an -dimensional -standard Wiener process;
(3) for all
(4)for almost all , is -measurable;
for any and .
We have the following.
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.
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.
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.
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.
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