Nonlinear Stochastic Evolution Problems in Applied Sciences
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Universita deg/i Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, Russia M. NIVAT, Universite de Paris VII, Paris, France
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands G.-C. ROTA, M.l.T., Cambridge, Mass., U.S.A.
Volume 82
Nonlinear Stochastic Evolution Problems in Applied Sciences
by
N. Bellomo
Department of Mathematics, Politecnico di Torino, Torino, Italy
Z. Brzezniak
Department of Mathematics, Cracowia University, Cracowia, Poland and
L. M. de Socio
Department of Mechanics,
University of Rome, "La Sapienza" , Rome,Italy
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library ofCongress Cataloging-in-Publication Data Bellomo. N.
Nonlinear stochastic evolution problems in applied sciences I N.
Bellomo. Z. Brzezniak. and L.M. de Soeio.
p. en. -- (Mathematlcs and its appl ieations ; v. 82) Ine ludes index.
ISBN 978-94-010-4803-3 ISBN 978-94-011-1820-0 (eBook) DOI 10.1007/978-94-011-1820-0
1. Stoehastic partial differentlal equations. 2. Oifferential equations. Nonlinear. I. Brzezniak. Z. II. De Soel0. L. M.
III. Tltle. IV. Series, Mathematies and its applicatlons (Kluwer Academic Publishers) ; v. 82.
OA274.25.B45 1993 519.2--dc20
ISBN 978-94-010-4803-3
Printed an acid-free paper
AII Rights Reserved
© 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 Softcover reprint of the hardcover 1 st edition 1992
92-35069
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
'Et moi, ...• si j'avait su comment en revenir, je n'y serais point aIle.'
Jules Verne The series is divergent; thererore we may be able to do something with it.
O. Heaviside
One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- sense'.
Eric T. Bell
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com- puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces_ And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of interac- tion areas one should add string theory where Riemann surfaces, algebraic geometry, modular func- tions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applica- ble. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accord- ingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on.
vi
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra mathematical sophistication that this requires. For that is where the rewards are. linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the non- linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci- ate· what I am hinting at: if electronics were linear we would have no fun with transistors and com- puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and u1trametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre- quently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading - fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub- series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis- cipline which are used in others. Thus the series still aims at books dealing with:
a central concept which plays an important role in several different mathematical and/or scientific speciaIization areas;
new applications of the results and ideas from one area of scientific endeavour into another;
influences which the results, problems and concepts of one field of enquiry have, and have had., on the development of another.
The shortest path between two truths in the rea! domain passes through the complex domain.
J. Hadamard
La physique ne nous donne pas seulement I' occasion de Rsoudre des problemes ... .ne nous fait prcssentir la solution.
H. Poincare Bussum, 1992
N ever lend books, for no one ever returns them; the only books I have in my library are books that other folk have lent me.
Anatole France
The function of an expert is not to be more right than other people, but to be wrong for more sophisticated reasons.
David Butler Michiel Hazewinkel
CONTENTS
Preface . . . xi
Chapter 1 Stochastic Models and Random Evolution Equations 1 1.1 Introduction . . . 1 1.2 A Classification of Partial Differential Equations 3 1.3 Function Spaces and the Definition of Solution 9 1.4 Stochastic Calculus of the Solution Process 12
1.5 Plan of the Book . . . 20
References to Chapter 1 22
Chapter 2 Deterministic Systems with Random Initial Conditions 25 2.1 Introduction . . . 25
2.2 Introduction to the Mathematical Problem 27
2.3 The Mathematical Method 29
2.4 Some Generalizations 39
2.5 Applications . . . . References to Chapter 2
Chapter 3 The Random Initial Boundary Value Problem 3.1 Introduction . . . .
3.2 Mathematical Modelling in Stochastic Mechanics 3.3 The Mathematical Method . . . .
49 60
63 63 64 70
viii _ _ _ _ _ _ _ _ _ _ _ _ NONLINEAR STOCHASTIC EVOLUTION PROBLEMS
3.4 Problems in the Half-Space and Problems in Several Space Variables 75 3.5 The Random Heat Equation . . . 77 3.6 Moving Boundary Problems in One Space Dimension 91
3.7 Final Discussion 93
References to Chapter 3 98
Chapter 4 Stochastic Systems with Additional Weighted Noise 101
4.1 Introduction . . . 101
4.2 The Mathematical Model 103
4.3 Examples of Stochastic Partial Differential Equations in Mechanics. 106
4.4 The Mathematical Method 11 7
4.5 Error Estimates 123
References to Chapter 4 130
Chapter 5 Time Evolution of the Probability Density 135 5.1 The First and Second Order Probability Densities 135 5.2 Deterministic Systems with Random Initial Conditions 137 5.3 Evolution of the Probability Density . . . . . 143 5.4 On the Continuous Interpolation and Approximation of the
Probability Density and Entropy Functions . . . 147 5.5 Systems with Random Initial-Boundary Conditions and Parameters 149
5.6 Application and Discussion 155
5.7 Some Conclusive Remarks 163
References to Chapter 5 . 164
Chapter 6 Some Further Developments of the SAl Method • 167 6.1 Introduction . . . 167 6.2 Systems of Coupled Partial and Ordinary Differential Equations 168 6.3 Integra-Differential Equations . . . 173
CONTENTS _____________________________________________________ ix
6.4 Ill-Posed Problems 6.5 Applications . 6.6 Final Remarks
References to Chapter 6 Appendix
Processes
Basic Concepts
References to Appendix . Authors Index
Subject Index
174 181 188 188
of Probability Theory and Stochastic 191 . 211
213 217
PREFACE
Physical phenomena of interest in science and technology are very often the- oretically simulated by means of models which correspond to partial differential equations. These equations are - in general - nonlinear and, as such, their solution is usually a difficult task. In this respect, linearization is possible only under rather stringent assumptions. In addition, the more realistic mathematical models show a random character. This last point can be quickly realized if one considers that, in practice, any system undergoes perturbations from the surrounding ambient and, therefore, the behaviour of the system itself is, in several circumstances, far away from the simple conditions of the ideal deterministic representation.
A further and even more important source of randomness for the mathematical models of real processes is represented by the effect of the so called "hidden"
variables. To explain this, one has just to think to the fact that, in order to manage with models of not extreme difficulty, not all the variables influencing a real phenomenon can be taken into account, but the state of the system is represented by a limited and little number of state parameters. It is this the way by which one tries to cope with the need of understanding the evolution of sometimes very complicated situations in physics. Since only the most important state variables are considered, the forgotten (hidden) ones still play their role as causes of a random behaviour of the model.
As a consequence of what we have discussed before. a realistic description of xi
xii _ _ _ _ _ _ _ _ _ _ _ _ _ _ NONLINEAR STOCHASTIC EVOLUTION PROBLEMS
the evolution of the state of a system is often given in terms of stochastic nonlinear differential problems.
Various methods are available in order to prove existence and uniqueness theorems related with some of these mathematical problems and several techniques have been proposed to get quantitative evaluations of the solutions.
This book is mostly concerned with the actual computation of the solution to nonlinear stochastic evolution problems governed by partial differential equations.
The aim is at modeling and solving rather than proving the existence of the solu- tions although these mathematical proofs are shown when this is necessary from the point of view of the applications.
The content of the book essentially deals with the applications of a "Stochastic Interpolation Method" which is based on the interpolation in space of the solution through the values which it takes in a number of selected nodal points. These values are to be determined via the solution of a finite set of ordinary differential equations which represent the evolution of the state variables in the nodes. The solution of an initial-boundary value problem is, in other words, achieved by solving a set of ordinary differential equations and then by interpolating the nodal values of the state parameters. In this process there are two main mathematical questions to be answered. One of them concerns the proof of the existence of the solutions in function spaces consistent with the application of the proposed method; the other one is how to estimate the distance between the solution obtained by the stochastic interpolation method and the one of the original partial differential equations. Both of these problems are dealt with in this book. In fact, apart from the solution technique which is the substance of the content, the book pays some attention also to some aspects of mathematical stochastic analysis.
Turning now to more details, each chapter contains, at least, one study case which can be a proof of existence or an application of the solution method or both.
PREFACE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ xiii
Very often, the problems are carried out in details and quantitative results are presented, the examples being taken from various fields of mathematical physics and applied sciences.
The content of the book is divided as follows:
Chapter 1 gives a general presentation of mathematical models in continuum mechanics and a description of the way by which problems are formulated. The related stochastic calculus of the solution is presented in order to indicate how the statistical properties of the solution itself can be practically evaluated. An application concerning a problem of gas-dynamics, in a case where the analytic solution is known, is an example of this stochastic calculus.
Chapter 2 deals with the "relatively easier" problem of the evolution of an unconstrained system with random space-dependent initial conditions, but which is governed by a deterministic evolution equation. In this case the method is applied to a simple class of problems (other chapters will present applications to more complicated mathematical situations). In Chapter 2 two mathematical models in fluid-dynamics are considered, the first one is of interest in the kinetic theory of gases and the other one is connected with the fluid-dynamics of continuum.
Chapter 3 deals with the initial-boundary value problem for equations with random initial and boundary conditions as well as with random parameters. The randomness is modelled by stochastic separable processes. The application con- cerns the solution of the random heat equation with either fixed or moving bound- aries and the actual computations of the solutions are pedormed in some signif- icant situations. The same chapter contains some calculation of the moments of the solution process for the non-linear hyperbolic equations of the vibrating string.
Chapter 4 is devoted to the initial-boundary value problem for models with additional noise, which obey to partial differential equations of Ito's type. Ito's solution method is adopted together with the stochastic interpolation procedure.
xiv _ _ _ _ _ _ _ _ _ _ _ _ _
NONUNEAR STOCHASTIC EVOLUTION PROBLEMS
With Chapter 4 the book enters into the analysis of genuinely stochastic systems.
In fact, the preceding chapters were essentially oriented to take the reader to the understanding of the transition from a deterministic system to a stochastic one in continuum fields.
Chapter 5 deals with the analysis of the time-evolution of the probability den- sity functions and of the related entropy functions connected with the dependent variables. Two applications in continuum physics are also presented. Therefore, Chapter 5 is essentially devoted to the qualitative and quantitative analysis of the chaotic behaviour of systems in continuum physics.
The final Chapter 6 provides indications on the solution of ill-posed and in- verse problems of the stochastic type and suggests guidelines for future research work in this not yet sufficiently explored field. Chapter 6 deals thus with problems which are surely of great relevance in applied technological and life sciences and are, at the same time, generally hard obstacles to confront with.
The Appendix gives a brief presentation of the theory of stochastic processes.
The book is at a postgraduate level and is addressed to applied mathemati- cians, engineers and scientists who work in technology and natural sciences and are interested in solving stochastic problems of physics and mechanics of continuum.
