Probabilistic Methods in Fluids This page intentionally left blank 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 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-PublicationD ata A catalogue record for this book is available from the British Library. PROBABILISTIC METHODS IN FLUIDS Proceedings of the Swansea 2002 Workshop Copyright 0 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. 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ISBN 981-238-226-7 Printed in Singapore by World Scientific Printers (S) Pte Ltd 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 ix