Partial differential equations in action: From modelling to theory

568 Pages·2010·3.919 MB·English

by  Sandro Salsa

To Anna, my wife
Sandro Salsa
Partial Differential
Equations in Action
From Modelling
to Theory
SandroSalsa
Dipartimentodi Matematica
Politecnico di Milano
CIP-Code:2007938891
ISBN978-88-470-0751-2 Springer Milan BerlinHeidelberg New York
e-ISBN978-88-470-0752-9
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Preface
Thisbookisdesignedasanadvancedundergraduateorafirst-yeargraduatecourse
forstudentsfromvariousdisciplineslikeappliedmathematics,physics,engineering.
Ithasevolvedwhileteachingcoursesonpartialdifferentialequations(PDE)during
the last few years at the Politecnico of Milan.
The main purpose of these courses was twofold: on the one hand, to train
the students to appreciate the interplay between theory and modelling in prob-
lems arising in the applied sciences, and on the other hand to give them a solid
theoretical background for numerical methods, such as finite elements.
Accordingly,this textbook is divided into two parts.
Thefirstone,chapters 2to5,hasarather elementarycharacter withthegoal
ofdevelopingandstudyingbasicproblemsfromthemacro-areasofdiffusion,prop-
agation and transport, waves and vibrations. I have tried to emphasize, whenever
possible,ideasandconnections withconcrete aspects, inorder toprovideintuition
and feeling for the subject.
Forthispart,aknowledgeofadvanced calculusandordinarydifferentialequa-
tions is required. Also, the repeated use of the method of separation of variables
assumessomebasicresultsfromthetheoryofFourierseries,whicharesummarized
in appendix A.
Chapter 2 starts with the heat equation and some of its variants in which
transport and reaction terms are incorporated. In addition to the classical top-
ics, I emphasized the connections with simple stochastic processes, such as ran-
domwalksandBrownianmotion.Thisrequires the knowledgeofsomeelementary
probability. It is my belief that it is worthwhile presenting this topic as early as
possible,evenatthepriceofgivinguptoalittlebitofrigorinthepresentation.An
application to financial mathematics shows the interaction between probabilistic
and deterministic modelling.The last two sections are devoted to two simple non
linear models from flow in porous medium and population dynamics.
Chapter 3 mainly treats the Laplace/Poisson equation. The main properties
of harmonic functions are presented once more emphasizing the probabilistic mo-
tivations. The second part of this chapter deals with representation formulas in
VI Preface
terms of potentials. In particular, the basic properties of the single and double
layer potentials are presented.
Chapter 4 is devoted to first order equations and in particular to first order
scalarconservationlaws.Themethods ofcharacteristics andthenotionofintegral
solution are developed through a simple model from traffic dynamics. In the last
part, the method of characteristics is extended to quasilinear and fully nonlinear
equations in two variables.
Inchapter 5thefundamentalaspects ofwavespropagationareexamined,lead-
ingtothe classicalformulasofd’Alembert,KirchhoffandPoisson.Inthe finalsec-
tion,theclassicalmodelforsurfacewavesindeepwaterillustratesthephenomenon
of dispersion, with the help of the method of stationary phase.
Themaintopicofthe second part,fromchapter6to9,isthedevelopmentof
Hilbert spaces methods for the variational formulation and the analysis of linear
boundary and initial-boundary value problems. Given the abstract nature of these
chapters, I have made an effort to provide intuition and motivation about the
variousconcepts andresults, runningtheriskofappearingabitwordysometimes.
The understanding of these topics requires some basic knowledge of Lebesgue
measure and integration,summarized in appendix B.
Chapter 6 contains the tools from functional analysis in Hilbert spaces, nec-
essary for a correct variational formulation of the most common boundary value
problems.Themainthemeisthesolvabilityofabstractvariationalproblems,lead-
ingtothe Lax-Milgramtheorem and Fredholm’salternative.Emphasis isgivento
the issues of compactness and weak convergence.
Chapter 7 is dividedintotwo parts.The first one is a briefintroduction tothe
theory of distributions of L. Schwartz. In the second one, the most used Sobolev
spaces and their basic properties are discussed.
Chapter 8 is devoted to the variationalformulationof elliptic boundary value
problemsandtheirsolvability.Thedevelopmentstartswithone-dimensionalprob-
lems,continues with Poisson’s equationand ends with general second order equa-
tions in divergence form. The last section contains an application to a simple
control problem, with both distributed observation and control.
The issue in chapter 9 is the variationalformulationof evolution problems, in
particularofinitial-boundaryvalueproblems forsecond order parabolicoperators
in divergence form and for the wave equation. Also, an application to a simple
control problem with final observation and distributed control is discussed.
At the end of each chapter, a number of exercises is included. Some of them
can be solved by a routine application of the theory or of the methods developed
in the text. Other problems are intended as a completion of some arguments or
proofs in the text. Also,there are problems in which the student is required to be
more autonomous. The most demanding problems are supplied with answers or
hints.
The order of presentation of the material is clearly a consequence of my ...
prejudices. However, the exposition ifflexibleenough to allowsubstantialchanges
Preface VII
withoutcompromisingthe comprehensionandtofacilitateaselection oftopicsfor
a one or two semester course.
Inthefirstpart,thechaptersareinpracticemutuallyindependent,withtheex-
ceptionofsubsections 3.3.6and3.3.7,whichpresumetheknowledgeofsection 2.6.
In the second part, which, in principle, may be presented independently of
the first one, more attention has to be paid to the order of the arguments. In
particular,the materialin chapter 6 and insections 7.1–7.4and 7.7–7.10is neces-
sary for understanding chapter 8, while chapter 9 uses concepts and results from
section 7.11.
Acknowledgments. While writing this book I benefitted from comments,
suggestions and criticisms of many collegues and students.
AmongmycolleguesIexpressmygratitudetoLucaDed´e,FaustoFerrari,Carlo
Pagani, Kevin Payne, Alfio Quarteroni, Fausto Saleri, Carlo Sgarra, Alessandro
Veneziani, Gianmaria A. Verzini and, in particular to Cristina Cerutti, Leonede
De Michele and Peter Laurence.
Among the students who have sat throuh my course on PDE, I would like to
thank Luca Bertagna, Michele Coti-Zelati, Alessandro Conca, Alessio Fumagalli,
Loredana Gaudio,Matteo Lesinigo, Andrea Manzoni and Lorenzo Tamellini.
Contents
Preface ............................................................ V
1 Introduction ................................................... 1
1.1 Mathematical Modelling...................................... 1
1.2 Partial Differential Equations ................................. 2
1.3 Well Posed Problems......................................... 5
1.4 Basic Notations and Facts .................................... 7
1.5 Smooth and Lipschitz Domains................................ 10
1.6 Integration by Parts Formulas................................. 11
2 Diffusion ....................................................... 13
2.1 The Diffusion Equation ...................................... 13
2.1.1 Introduction .......................................... 13
2.1.2 The conduction of heat ................................ 14
2.1.3 Wellposed problems (n=1)............................ 16
2.1.4 A solution by separation of variables..................... 19
2.1.5 Problems in dimension n>1 ........................... 27
2.2 Uniqueness ................................................. 30
2.2.1 Integral method....................................... 30
2.2.2 Maximumprinciples ................................... 31
2.3 The Fundamental Solution.................................... 34
2.3.1 Invarianttransformations............................... 34
2.3.2 Fundamentalsolution (n=1)........................... 36
2.3.3 The Dirac distribution ................................. 39
2.3.4 Fundamentalsolution (n>1)........................... 42
2.4 Symmetric RandomWalk(n=1) ............................. 43
2.4.1 Preliminarycomputations .............................. 44
2.4.2 The limittransition probability ......................... 47
2.4.3 From random walk to Brownian motion .................. 49
2.5 Diffusion,Drift and Reaction ................................. 52
2.5.1 Randomwalk with drift................................ 52
X Contents
2.5.2 Pollutionin a channel.................................. 54
2.5.3 Randomwalk with drift and reaction .................... 57
2.6 MultidimensionalRandomWalk............................... 58
2.6.1 The symmetric case ................................... 58
2.6.2 Walkswith drift and reaction ........................... 62
2.7 An Example of Reaction−Diffusion(n=3) ..................... 62
2.8 The GlobalCauchy Problem (n=1)........................... 68
2.8.1 The homogeneous case ................................. 68
2.8.2 Existence of a solution ................................. 69
2.8.3 The non homogeneous case. Duhamel’smethod ........... 71
2.8.4 Maximumprinciples and uniqueness ..................... 74
2.9 An Application to Finance.................................... 77
2.9.1 European options ..................................... 77
2.9.2 An evolutionmodel for the price S ...................... 77
2.9.3 The Black-Scholes equation............................. 80
2.9.4 The solutions ......................................... 83
2.9.5 Hedging and self-financing strategy ...................... 88
2.10 Some Nonlinear Aspects...................................... 90
2.10.1 Nonlinear diffusion. The porous medium equation ......... 90
2.10.2 Nonlinear reaction. Fischer’s equation.................... 93
Problems ....................................................... 97
3 The Laplace Equation ..........................................102
3.1 Introduction ................................................102
3.2 Well Posed Problems. Uniqueness .............................103
3.3 Harmonic Functions .........................................105
3.3.1 Discrete harmonic functions ............................105
3.3.2 Mean value properties .................................109
3.3.3 Maximumprinciples ...................................110
3.3.4 The Dirichlet problem in a circle. Poisson’s formula........113
3.3.5 Harnack’s inequality and Liouville’stheorem..............117
3.3.6 A probabilisticsolution of the Dirichlet problem...........118
3.3.7 Recurrence and Brownian motion .......................122
3.4 Fundamental Solutionand Newtonian Potential .................124
3.4.1 The fundamental solution ..............................124
3.4.2 The Newtonian potential...............................126
3.4.3 A divergence-curl system.
Helmholtz decomposition formula .......................128
3.5 The Green Function .........................................132
3.5.1 An integral identity....................................132
3.5.2 The Green function....................................133
3.5.3 Green’s representation formula..........................135
3.5.4 The Neumann function.................................137
3.6 Uniqueness in Unbounded Domains............................139
3.6.1 Exterior problems .....................................139
Contents XI
3.7 Surface Potentials ...........................................141
3.7.1 The double and single layer potentials ...................142
3.7.2 The integral equations of potential theory ................146
Problems .......................................................150
4 Scalar Conservation Laws and First Order Equations...........156
4.1 Introduction ................................................156
4.2 Linear Transport Equation ...................................157
4.2.1 Pollutionin a channel..................................157
4.2.2 Distributed source.....................................159
4.2.3 Decay and localized source .............................160
4.2.4 Inflow and outflow characteristics. A stabilityestimate .....162
4.3 Traffic Dynamics ............................................164
4.3.1 A macroscopic model ..................................164
4.3.2 The method of characteristics...........................165
4.3.3 The green lightproblem................................168
4.3.4 Traffic jam ahead......................................172
4.4 Integral (or Weak) Solutions ..................................174
4.4.1 The method of characteristics revisited...................174
4.4.2 Definition of integral solution ...........................177
4.4.3 The Rankine-Hugoniotcondition ........................179
4.4.4 The entropy condition .................................183
4.4.5 The Riemann problem .................................185
4.4.6 Vanishingviscosity method .............................186
4.4.7 The viscous Burger equation............................189
4.5 The Method of Characteristics for Quasilinear Equations .........192
4.5.1 Characteristics ........................................192
4.5.2 The Cauchy problem ..................................194
4.5.3 Lagrange method of first integrals .......................202
4.5.4 Underground flow .....................................205
4.6 General First Order Equations ................................207
4.6.1 Characteristic strips ...................................207
4.6.2 The Cauchy Problem ..................................210
Problems .......................................................214
5 Waves and Vibrations..........................................221
5.1 General Concepts............................................221
5.1.1 Types of waves........................................221
5.1.2 Group velocity and dispersion relation ...................223
5.2 Transversal Waves in a String.................................226
5.2.1 The model ...........................................226
5.2.2 Energy...............................................228
5.3 The One-dimensional Wave Equation ..........................229
5.3.1 Initialand boundary conditions .........................229
5.3.2 Separation of variables .................................231