Partial Differential Equations in Action: From Modelling to Theory

699 Pages·2016·5.4 MB·English

by  Sandro Salsa

UNITEXT – La Matematica per il 3+2
Volume 99
Editor-in-chief
A.Quarteroni
Serieseditors
L.Ambrosio
P.Biscari
C.Ciliberto
M.Ledoux
W.J.Runggaldier
Moreinformationaboutthisseriesathttp://www.springer.com/series/5418
Sandro Salsa
Partial Differential
Equations in Action
From Modelling to Theory
Third Edition
SandroSalsa
DipartimentodiMatematica
PolitecnicodiMilano
Milano,Italy
ISSN2038-5722 ISSN2038-5757(electronic)
UNITEXT–LaMatematicaperil3+2
ISBN978-3-319-31237-8 ISBN978-3-319-31238-5(eBook)
DOI10.1007/978-3-319-31238-5
LibraryofCongressControlNumber:2016932390
©SpringerInternationalPublishingSwitzerland2016
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ToAnna, my wife
Preface
Thisbookisdesignedasanadvancedundergraduateorafirst-yeargraduatecourse
for students from various disciplines like applied mathematics, physics, engineer-
ing.It has evolvedwhile teaching courses onpartialdifferential equations (PDEs)
during the last few years at the Politecnico di Milano.
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 problems
arising in the applied sciences, and on the other hand to give them a solid theo-
retical background for numerical methods, such as finite elements.
Accordingly, this textbook is divided into two parts, that we briefly describe
below, writing in italics the relevant differences with the first edition, the second
one being pretty similar.
The first part,Chaps.2to 5,has arather elementary character withthe goal
ofdevelopingandstudyingbasicproblemsfromthemacro-areasofdiffusion,prop-
agationand transport, waves and vibrations. I have tried to emphasize, whenever
possible,ideasandconnections withconcrete aspects, inordertoprovideintuition
and feeling for the subject.
Forthispart, aknowledgeofadvanced calculus andordinary differentialequa-
tions is required. Also, the repeated use of the method of separation of variables
assumessomebasicresultsfromthetheoryofFourierseries,whicharesummarized
in Appendix A.
Chapter2startswiththeheatequationandsomeofitsvariantsinwhichtrans-
portandreaction terms are incorporated. Inadditiontothe classical topics, Iem-
phasized the connections with simple stochastic processes, such as random walks
and Brownian motion. This requires the knowledge of some elementary probabil-
ity. It is my belief that it is worthwhile presenting this topic as early as possible,
even at the price of givingup to a little bit of rigor in the presentation. An appli-
cation to financial mathematics shows the interaction between probabilistic and
deterministicmodelling.Thelasttwosectionsaredevotedtotwosimplenonlinear
models from flow in porous medium and population dynamics.
viii Preface
Chapter 3mainlytreats the Laplace/Poisson equation.The mainproperties of
harmonic functions are presented once more emphasizing the probabilistic moti-
vations. I have included Perron’s method of sub/super solution, due to is renewed
importanceasa solutiontechnique forfullynon linearequations.The second part
ofthischapter dealswithrepresentation formulasinterms ofpotentials.Inpartic-
ular,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 or-
der scalar conservation laws. The methods of characteristics and the notions of
shock and rarefaction waves are introduced through a simple model from traffic
dynamics. An application to sedimentation theory illustrates the method for non
convex/concave flux function. In the last part, the method of characteristics is
extended to quasilinear and fully nonlinear equations in two variables.
In Chap. 5 the fundamental aspects of waves propagation are examined, lead-
ingtothe classical formulasofd’Alembert,KirchhoffandPoisson.A simple model
of Acoustic Thermography serves as an application of Huygens principle. In the
final section, the classical model for surface waves in deep water illustrates the
phenomenon of dispersion, with the help of the method of stationary phase.
The second partincludes the twonew Chaps. 9 and 11. In Chaps.6to10we
developeHilbertspacesmethodsforthevariationalformulationandtheanalysisof
mainly linear boundary and initial-boundary value problems. Given the abstract
natureofthesechapters, Ihavemadeanefforttoprovideintuitionandmotivation
about the various concepts and results, sometimes running the risk of appearing
a bit wordy. 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-
ingto the Lax-Milgramtheorem andFredholm’salternative.Emphasisis givento
the issues of compactness and weak convergence. Section 6.10 is devoted to the
fixed point theorems of Banach and of Schauder and Leray-Schauder.
Chapter 7 is divided into two parts. The first one is a brief introduction to
thetheory ofdistributions(orgeneralized functions)ofL.Schwartz.Inthe second
one, the most used Sobolev spaces and their basic properties are discussed.
Chapter 8 is devoted to the variational formulation of linear elliptic bound-
ary value problems and their solvability. The development starts with Poisson’s
equation and ends with general second order equations in divergence form.
InChap.9Ihavegatheredanumberofapplicationsofthevariationaltheoryof
ellipticequations,inparticularto elastostaticsand to thestationary Navier-Stokes
equations. Also, an application to a simple control problem is discussed.
TheissueinChap.10,whichhasbeen almost completelyremodeled, isthevari-
ational formulation of evolution problems, in particular of initial-boundary value
problemsforsecond orderparabolicoperators indivergence formandforthewave
equation.
Preface ix
Chapter 11 contains abrief introduction to the basic concepts ofthe theory of
systems of first order conservation laws, in one spatial dimension. In particular
we extend from the scalar case of Chap 4, the notions of characteristics, shocks,
rarefaction waves, contact discontinuity and entropy condition.The mainfocus is
the solution of the Riemann problem.
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 inthe 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.
Other(completelysolved)exercisescanbefoundin[17],thenaturalcompanion
of this book by S. Salsa, G. Verzini, Springer 2015.
The order of presentation of the material is clearly a consequence of my ...
prejudices. However, the exposition ifflexibleenough to allowsubstantialchanges
withoutcompromisingthecomprehension andtofacilitateaselectionoftopicsfor
a one or two semester course.
In the first part, the chapters are, in practice, mutually independent, with
the exception of Subsection 3.3.1 and Sect. 3.4, which presume the knowledge of
Sect. 2.6.
Inthesecondpart,moreattentionhastobepaidtotheorderofthearguments.
The material in Sects. 6.1–6.9 and in Sect. 7.1–7.4 and 7.7–7.10 is necessary for
understandingthetopicsinChap.8–10.Moreover,Chap.9requiresalsoSect.6.10,
while to cover Chap. 10, also concepts and results from Sect. 7.11 are needed.
Finally,Chap. 11 uses Subsections 4.4.2,4.4.3and 4.6.1.
Acknowledgments.While writingthis book,during the first edition, I bene-
fitted from comments and suggestions of many collegues and students.
Amongmycollegues,IexpressmygratitudetoLucaDedé,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.
Amongthe students whohavesat through mycourse onPDEs, Iwouldliketo
thank Luca Bertagna, Michele Coti-Zelati, Alessandro Conca, Alessio Fumagalli,
Loredana Gaudio, Matteo Lesinigo, Andrea Manzoni and Lorenzo Tamellini.
Fo the last two editions, I am particularly indebted to Leonede de Michele,
Ugo Gianazza and Gianmaria Verzini for their interest, criticism and contribu-
tion.ManythanksgotoMichele DiCristo, GiovanniMolica-Bisci,NicolaParolini
AttilioRaoandFrancescoTulonefortheircommentsandthetimewespentinpre-
cious (for me) discussions. Finally,I like to express my appreciation to Francesca
Bonadei and Francesca Ferrari of Springer Italia, for their constant collaboration
and support.
Milan,April 2016 Sandro Salsa
Contents
1 Introduction................................................... 1
1.1 Mathematical Modelling...................................... 1
1.2 Partial Differential Equations ................................. 2
1.3 WellPosed Problems......................................... 5
1.4 Basic Notations and Facts .................................... 7
1.5 Smooth and Lipschitz Domains................................ 12
1.6 Integration by Parts Formulas................................. 15
2 Diffusion....................................................... 17
2.1 The Diffusion Equation ...................................... 17
2.1.1 Introduction .......................................... 17
2.1.2 The conduction of heat................................. 18
2.1.3 Wellposed problems (n=1)............................ 20
2.1.4 A solution by separation of variables..................... 23
2.1.5 Problems in dimension n>1 ........................... 32
2.2 Uniqueness and MaximumPrinciples........................... 34
2.2.1 Integral method....................................... 34
2.2.2 Maximumprinciples ................................... 36
2.3 The Fundamental Solution.................................... 39
2.3.1 Invarianttransformations............................... 39
2.3.2 The fundamental solution (n=1) ....................... 41
2.3.3 The Dirac distribution ................................. 43
2.3.4 The fundamental solution (n>1) ....................... 47
2.4 Symmetric RandomWalk (n=1) ............................. 48
2.4.1 Preliminarycomputations .............................. 49
2.4.2 The limittransition probability ......................... 52
2.4.3 From random walk to Brownian motion .................. 54
2.5 Diffusion, Drift and Reaction ................................. 58
2.5.1 Randomwalk with drift................................ 58
2.5.2 Pollutionin a channel.................................. 60
2.5.3 Randomwalk with drift and reaction .................... 63