Some results from Probability Theory and Stochastic Calculus

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Introduction

The study of turbulence either in Newtonian fluids or Non-Newtonian fluids is one of the greatest unsolved and still not well understood problems in contemporary applied sciences. For indepth 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 [48, 80, 88, 110] and references therein. It is also a commonly accepted fact that the rigorous understanding of turbulence is one of the most challenging task for the future development of certain fields of mathematics such as analysis and theory of partial differential equations. 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, makes the problem more realistic. Such approach in the understanding of the turbulence phenomenon was pioneered by Bensoussan and Temam in [10] where they studied the stochastic Navier-Stokes equations (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, for instance, to [2], [8], [16], [19], [37], [38], [44], [83], [95],[98], [101], [105], [106]. Similar investigations for Non-Newtonian fluids have almost not been undertaken except in very few work; we refer, for instance, to [59],[66], [67], [78], [92], [118] for some computational studies of stochastic models of polymeric fluids and to [14], [65], [68], [69] for their mathematical analysis.

Some results from Probability Theory and Stochastic Calculus

In this section, we give some basic definitions and classical theorems from Probability Theory and Stochastic Analysis. We do not provide too much details since most of them are very well-known. For the details and for further reading on Probability Theory and Stochastic Analysis, we urge the reader to consult [4], [36], [53], [70], [71], [95], [96], [103], [111] among many other references. Let (Ω, F, P) be a probability space, where Ω is a set (it may be a topological vector space) with elements ω, F denotes the Borel σ-field of subsets of Ω, and P is a probability measure. Throughout we denote by E the mathematical expectation associated to the probability measure P. Definition 2.9. Let (E, E) be a measurable set. Any measurable mapping X : Ω → E is called E-valued random variable or a random variable in E. Let T > 0 and I = [0, T], a stochastic process in E is any family Xt = (X(t), t ∈ I)) of random variables in E. It is said continuous if its sample paths Xt(ω) or X(t, ω) is a continuous function of t for almost all (almost everywhere) ω ∈ Ω. A process Yt is a modification or a version of Xt if P(ω : Xt(ω) = Yt(ω)) = 1, ∀t ∈ I. Throughout this work we will make no difference between Xt and its version. The following theorem is a simple criterion for the existence of a continuous version of a realvalued process Xt . We refer to [71, 103] for its proof and some of its extensions. Theorem 2.10 (Kolmogorov-Centsov) ˘ . Suppose that a real-valued process X = {Xt , 0 ≤ t ≤ T} on a probability space (Ω, P) satisfies the condition E|Xt+h − Xt | γ ≤ Ch1+β , 0 ≤ t, h ≤ T, for some positive constants γ, β, and C. Then there exists a continuous modification X˜ = {X˜ t , 0 ≤ t ≤ T} of X, which is locally H¨older-continuous with exponent κ ∈ (0, β γ ). A filtration (F t )0≤t≤T is an increasing σ-fields F t ⊂ F, t ∈ I.

On the uniqueness of the strong probabilistic solution

As already mentioned in the introduction we also discuss the pathwise uniqueness of solution. More precisely we prove the following result. Theorem 3.12. Let u1 and u2 be two strong solutions defined on the filtered probability space (Ω, F, F t , P) of the problem (1.5). If we set v = u1 − u2, then we have v = 0 almost surely. Proof. Let u1 and u2 be two solutions with the same initial condition u0 and let v = u1−u2. It can be shown that the process v satisfies the equation d(v(t) + αAv(t)) + νAv(t)dt − F(u1(t), t) − F(u2(t), t)dt + G(u1(t), t) − G(u2(t), t)dW = −P(curl(u1(t) − α∆u1(t)) × u1(t) − curl(u2(t) − α∆u2(t)) × u2(t))dt, CHAPTER 3. STRONG PROBABILISTIC SOLUTION OF THE MODEL 50 and v(0) = 0. We obtain by multiplying this equation by (I + αA) −1 and by applying Itˆo’s formula to |v| 2 V (see [73] or [94] for the infinite dimensional version of Theorem 2.14) that |v(t)| 2 V + 2 Z t 0 ν||v(s)||2 − (curl(v(s) − α∆v(s)) × v(s), u2(s)) ds = Z t 0 2(Fb(u1(s), s) − Fb(u2(s), s), v(s))V + |Gb(u1(s), s) − Gb(u2(s), s)| 2 V⊗m ds + 2 Z t 0 (Gb(u1(s), s) − Gb(u2(s), s), v(s))VdW.

Analysis of the asymptotic behavior of the solutions

In this last section we establish some results on the long time behavior of the strong solutions of the stochastic model for the second grade fluids. We consider two subsections. In the first, we study the exponential decay in mean square of the strong solution. In the second, we strengthen the assumption on the external force F and investigate the exponential stability in mean square of the non-trivial stationary solution.

Existence of weak probabilistic solutions

In many cases of interest the Lipschitz condition on the forcing terms F(v, t) and G(v, t) no longer holds. In this chapter we consider such situation. More specifically, we suppose that F(v, t) ang G(v, t) are only continuous with respect to the variable v. The appropriate notion solution in this case is that of weak probabilistic solutions known as well as martingale solutions. Here we prove the existence of such solutions. To do so we mainly use a compactness method which seems to have been initially introduced by Bensoussn [8], [7]. In contrast to most work dealing with martingale solutions of SPDEs ([20], [17], [25], [43], [44], [72]…), we do not use the martingale representation. The current chapter is organized as follows. Section 2 is devoted to the formulation of the hypotheses and the main result. We introduce a Galerkin approximation of the problem and derive crucial a priori estimates for its solutions in Section 3; a compactness result is also derived. We prove the main result in Section 4.

Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids

This chapter is devoted to the analysis of the behavior of the solutions of the stochastic equations for the motion of turbulent flows of a second grade fluids when the stress modulus α tends to zero. More precisely for a periodic square D, D = [0, L] 2 ⊂ R 2 , L > 0, we aim to study the convergence of the periodic-in-space velocity solution with period L of the following: when the parameter α tends to 0. Here P is a scalar function representing a modified pressure, (Ω, F, P), 0 ≤ t ≤ T, is a given complete probability space on which a R mvalued standard Wiener process W is defined and F t is an increasing filtration generated 79 CHAPTER 5. ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS AS α → 0 80 by W. Our aim is to show that we can construct a sequence u αj of strong probabilistic solutions of (5.1) that converges in appropriate sense the strong probabilistic solution of the stochastic Navier-Stokes equations as αj → 0. That is, there exists another complete filtered probability space (Ω¯, F¯, F¯t , P¯), a R m-valued Wiener process W¯ and a stochastic process v such that the following holds in the distribution sense.

Contents :

• Abstract
• Acknowledgements
• Contents
• 1 Introduction
  • 1.1 Physical background of second grade fluids
  • 1.2 Overview of the thesis
• 2 Preliminary Results
  • 2.1 Analytical preliminaries
  • 2.2 Some results from Probability Theory and Stochastic Calculus
• 3 Strong probabilistic solution of the model
  • 3.1 Introduction
  • 3.2 Existence of the strong probabilistic solution
  • 3.2.1 Hypotheses and statement of the existence theorem
  • 3.2.2 Proof of the existence result
  • 3.3 On the uniqueness of the strong probabilistic solution
  • 3.4 Analysis of the asymptotic behavior of the solutions
  • 3.4.1 The exponential decay of the strong probabilistic solution
  • 3.4.2 The stability of the stationary solution
• 4 Existence of weak probabilistic solutions
  • 4.1 Introduction
  • 4.2 Hypotheses and the main result
  • 4.2.1 Hypotheses
  • 4.2.2 Statement of the existence theorem of weak probabilistic solutions
  • 4.3 Auxiliary results
  • 4.3.1 The approximate solution
  • 4.3.2 A priori estimates
  • 4.3.3 Tightness property and application of Prokhorov’s and Skorokhod’s
  • theorems
  • 4.4 Proof of the main result
  • 4.4.1 Passage to the limit
  • 4.4.2 Pathwise continuity in time of the weak probabilistic solution
• 5 Asymptotic behavior of the solutions as α →
  • 5.1 Introduction
  • 5.2 Hypotheses and a convergence theorem
  • 5.3 Uniform a priori estimates
  • 5.4 Proof of Theorem
• Conclusion
• Bibliography

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Some mathematical problems in the dynamics of stochastic second-grade fluids