Table Of ContentProbabilistic
Methods in
Fluids
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editors:
I M Davies
N Jacob
A Truman
Department of Mathematics
University of Wales Swansea
UK
0 Hassan
K Morgan
N P Weatherill
School of Engineering
University of Wales Swansea
Proceedings of the Swansea 2002 Workshop
Probabilistic
Methods in
Fluids
Wales, UK 14 - 19 April 2002
ye World Scientific
L NewJersey London Singapore Hong Kong
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PROBABILISTIC METHODS IN FLUIDS
Proceedings of the Swansea 2002 Workshop
Copyright 0 2003 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means,
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
IRIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Participants . . . . . . . . . . . . . . . . . . . . . . . . . xi
Sergio Albeverio and Yana Belopolskaya . . . . . . . . . . . . . 1
Probabilistic Approach to Hydrodynamic Equations
Hakima Bessaih and Franco Flandoli . . . . . . . . . . . . . . 22
A Mean Field Result for 3D Vortex Filaments
Bjorn Bottcher and Niels Jacob . . . . . . . . . . . . . . . . . 35
Remarks on Meixner-type Processes
Zdzistaw Brzeinaak . . . . . . . . . . . . . . . . . . . . . . 48
Some Remarks on It6 and Stratonovich Integration in 2-smooth
Banach Spaces
Tomas Caraballo . . . . . . . . . . . . . . . . . . . . . . . 70
The Long-time Behaviour of Stochastic 2D-Navier-Stokes Equations
Pao-Liu Chow . . . . . . . . . . . . . . . . . . . . . . . . 84
Semilinear Stochastic Wave Equations
Nigel J. Cutland . . . . . . . . . . . . . . . . . . . . . . . 97
Stochastic Navier-Stokes Equations: Loeb Space
Techniques & Attractors
Arnaud Debussche . . . . . . . . . . . . . . . . . . . . . 115
The 2D-Navier-Stokes Equations Perturbed by a Delta
Correlated Noise
Sergio Albeverio and Benedetta Ferrario . . . . . . . . . . . . 130
Invariant Measures of Lkvy-Khinchine Type for 2D Fluids
Franco Flandoli . . . . . . . . . . . . . . . . . . . . . . . 144
Some Remarks on a Statistical Theory of Turbulent Flows
Christophe Giraud . . . . . . . . . . . . . . . . . . . . . 161
Some Properties of Burgers Turbulence with White Noise
Initial Conditions
V
vi
Yuri E. Gliklikh . . . . . . . . . . . . . . . . . . . . . . . 179
Deterministic Viscous Hydrodynamics via Stochastic Processes on
Groups of Diffeomorphisms
Niels Jacob and Aubrey Truman . . . . . . . . . . . . . . . 191
Further Classes of Pseudo-differential Operators Applicable
to Modelling in Finance and Turbulence
Benjamin Jourdain and Tony Lelihre . . . . . . . . . . . . . 205
Mathematical Analysis of a Stochastic Differential Equation Arising
in the Micro-Macro Modelling of Polymeric Fluids
Hannelore Lisei and Michael Scheutzow . . . . . . . . . . . . 224
On the Dispersion of Sets under the Action of an Isotropic
Brownian Flow
Aubrey Truman, Chris N . Reynolds and David Williams . . . . . 239
Stochastic Burgers Equation in d-dimensions - A One-dimensional
Analysis: Hot and Cool Caustics and Intermittence
of Stochastic Turbulence
A m e n Shirikyan . . . . . . . . . . . . . . . . . . . . . . 263
A Version of the Law of Large Numbers and Applications
Maricin SlodiEka . . . . . . . . . . . . . . . . . . . . . . 272
Comprehensive Models for Wells
Enrique Thomann and Mina Ossiander . . . . . . . . . . . . 287
Stochastic Cascades Applied to the Navier-Stokes Equations
Aubrey Truman and Jiang-Lun Wu . . . . . . . . . . . . . . 298
Stochastic Burgers Equation with Lkvy Space-Time White Noise
TushengZhang . . . . . . . . . . . . . . . . . . . . . . . 324
A Comparison Theorem for Solutions of Backward Stochastic
Differential Equations with Two Reflecting Barriers and
Its Applications
Aubrey Truman and Huaizhong Zhao . . . . . . . . . . . . . 332
Burgers Equation and the WKB-Langer Asymptotic L2
Approximation of Eigenfunctions and Their Derivatives
Preface
This volume contains papers presented at the “Probabilistic Methods in
Fluids Workshop” which was hosted by the Department of Mathematics,
University of Wales Swansea between the 14th and l g t h of April 2002.
The aim of the meeting, the first IRIMA workshop, was to bring together
internationally reknowned researchers from the areas of Pure Mathemat-
ics, Applied Mathematics and Engineering to participate in a workshop,
on probabilistic methods for fluids, and through collaboration further the
mathematical understanding of the fundamental problems in this field.
This international workshop successfully allowed leading researchers to
present, reflect upon and discuss their recent work in the probabilistic
modelling of fluids. This field stretches across Pure Mathematics, Applied
Mathematics and Engineering and consequently is ideally placed to benefit
from regularly arranged workshops for collaborative purposes. The Work-
shop mainly concentrated on the understanding of turbulence in stochastic
fluid dynamics, a problem which has numerous applications in science and
engineering and has defied many attempts to success full^ model it. The
workshop bridged a gap between the recent year of activity at the Univer-
sity of Warwick and the year of emphasis at Princeton, which started in
Autumn 2002. As such the workshop ensured that the research momentum
in Britain, in this subject, was maintained.
In this volume probabilistic approaches to hydrodynamic equations are re-
viewed and deterministic viscous hydrodynamics is discussed in terms of
stochastic processes on groups of diffeomorphisms. At the Workshop sig-
nificant progress was made in understanding the intermittence of stochastic
turbulence for Burgers equation and the application of L6vy processes to
the Mathematics of Finance, both of which are represented in the pro-
ceedings. Other noteworthy developments concerned the Strong Law of
Large Numbers and ergodicity of the Gaussian invariant measures for 2-
dimensional Navier-Stokes equations with space-time white noise and pe-
riodic boundary conditions and mean field results for 3-dimensional vortex
filaments. Also, new results are presented on Loeb space techniques and
attractors for stochastic Navier-Stokes equations. The long time behaviour
of stochastic 2-dimensional Navier-Stokes equations is investigated as are
vii
...
Vll l
perturbations by delta correlated noise. Burgers turbulence for white noise
initial conditions is discussed in detail and the Cauchy problem for stochas-
tic Burgers equation with L6vy space-time white noise is also examined. A
complete mathematical analysis of stochastic differential equations arising
in micro-macro modelling of polymeric fluids is given.
Scientific Organising Committee S. Albeverio, Y.I. Belapolskaya, Z.
Brzezniak, A. Chorin, F. Flandoli, B. Rozovski, A. Truman
We are especially grateful to Zdzislaw Brzezniak for his contribution to
the organisation and success of the workshop, and to Roger Tribe for his
guidance.
finding
The workshop was supported by EPSRC grant GR/96545/01 “Probabil-
isitic Methods for Fluids - IRIMA” and we are indebted to EPSRC for
their financial support and advice.
Local Organisation
We wish to thank Jane Barham and Janice Lewis for their forebearance
before, during and after the workshop in providing secretarial and adminis-
trative support. We must thank also Bjorn Boettcher, Victoriya Knopova,
Scott Reasons and Chris Reynolds for their contribution towards the suc-
cessful running of the workshop.
Finally, we thank the referees for their important but anonymous contribu-
tion in helping us to finish this volume on time.
I M Davies 0 Hassan
N Jacob K Morgan
A Truman N P Weatherill
University of Wales Swansea, December 2002
International Research Institute in Mathematics and its
Applications
Patron: Sir Michael Atiyah OM, FRS
There is no such thing as Applied Science only the Applications of
Science, Henri Poincark
We have established the International Research Institute in Mathematics
and Its Applications, IRIMA, (Sefydliad Ymchwil Rhyngwladol i Fathe-
mateg a’i Chymwysiadau, SYRIFAC) with the aim of conducting a series
of research programmes in Mathematics and its applications to Engineer-
ing and Science. In so doing we aim to accelerate the transfer of modern
Mathematics to Engineering and the Sciences. These programmes should
be seen to be interdisciplinary, with the express intention of providing a
forum for interaction between groups of mathematicians, engineers and sci-
entists, while at the same time preserving the integrity of the Mathematics
being utilised.
The Institute will be based in Swansea and will draw on existing strengths in
Stochastic Processes, Physical Mathematics, Finite Element Methods and
Theoretical Computer Science. Swansea (in the person of Oleg Zienkiewicz
FRS) pioneered the use of Finite Element Methods in Engineering. More
recently, his research group, which includes Profs. Nigel Weatherill, Ken
Morgan FREng and Roger Owen FREng, has done vitally important re-
search work in a number of different application areas, including work on
the European Airbus and Thrust SSC, the supersonic car. Swansea also
has international centres of research excellence in Probability Theory and
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