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Appendix C. Systems of Ordinary Differential Equations

Appendix C. Systems of Ordinary Differential Equations

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PartialDifferentialEquations



PartialDifferentialEquations

AnIntroductiontoTheoryandApplications

MichaelShearer

RachelLevy

PRINCETONUNIVERSITYPRESS

PrincetonandOxford



Copyright©2015byPrincetonUniversityPress

PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540

IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock,OxfordshireOX201TW

press.princeton.edu

CoverphotographcourtesyofMichaelShearerandRachelLevy.

CoverdesignbyLorraineBetzDoneker.

AllRightsReserved

LibraryofCongressCataloging-in-PublicationData

Shearer,Michael.

Partialdifferentialequations:anintroductiontotheoryandapplications/MichaelShearer,RachelLevy.

Pagescm

Includesbibliographicalreferencesandindex.

ISBN978-0-691-16129-7(cloth:alk.paper)—ISBN0-691-16129-1(cloth:alk.paper)

1.Differentialequations,Partial.I.Levy,Rachel,1968–II.Title.

QA374.S452015

515′.353—dc23

2014034777

BritishLibraryCataloging-in-PublicationDataisavailable

ThisbookhasbeencomposedinMinionProwithMyriadProandDINdisplayusingZzTEXbyPrinceton

EditorialAssociatesInc.,Scottsdale,Arizona

Printedonacid-freepaper.

PrintedintheUnitedStatesofAmerica

10987654321



Contents

Prefaceix

1.Introduction1

1.1.LinearPDE2

1.2.Solutions;InitialandBoundaryConditions3

1.3.NonlinearPDE4

1.4.BeginningExampleswithExplicitWave-likeSolutions6

Problems8

2.Beginnings11

2.1.FourFundamentalIssuesinPDETheory11

2.2.ClassificationofSecond-OrderPDE12

2.3.InitialValueProblemsandtheCauchy-KovalevskayaTheorem17

2.4.PDEfromBalanceLaws21

Problems26

3.First-OrderPDE29

3.1.TheMethodofCharacteristicsforInitialValueProblems29

3.2.TheMethodofCharacteristicsforCauchyProblemsinTwoVariables32

3.3.TheMethodofCharacteristicsinRn35

3.4.ScalarConservationLawsandtheFormationofShocks38

Problems40

4.TheWaveEquation43

4.1.TheWaveEquationinElasticity43

4.2.D’Alembert’sSolution48

4.3.TheEnergyE(t)andUniquenessofSolutions56

4.4.Duhamel’sPrinciplefortheInhomogeneousWaveEquation57

4.5.TheWaveEquationonR2andR359

Problems61

5.TheHeatEquation65

5.1.TheFundamentalSolution66

5.2.TheCauchyProblemfortheHeatEquation68

5.3.TheEnergyMethod73



5.4.TheMaximumPrinciple75

5.5.Duhamel’sPrinciplefortheInhomogeneousHeatEquation77

Problems78

6.SeparationofVariablesandFourierSeries81

6.1.FourierSeries81

6.2.SeparationofVariablesfortheHeatEquation82

6.3.SeparationofVariablesfortheWaveEquation91

6.4.SeparationofVariablesforaNonlinearHeatEquation93

6.5.TheBeamEquation94

Problems96

7.EigenfunctionsandConvergenceofFourierSeries99

7.1.EigenfunctionsforODE99

7.2.ConvergenceandCompleteness102

7.3.PointwiseConvergenceofFourierSeries105

7.4.UniformConvergenceofFourierSeries108

7.5.ConvergenceinL2110

7.6.FourierTransform114

Problems117

8.Laplace’sEquationandPoisson’sEquation119

8.1.TheFundamentalSolution119

8.2.SolvingPoisson’sEquationinRn120

8.3.PropertiesofHarmonicFunctions122

8.4.SeparationofVariablesforLaplace’sEquation125

Problems130

9.Green’sFunctionsandDistributions133

9.1.BoundaryValueProblems133

9.2.TestFunctionsandDistributions136

9.3.Green’sFunctions144

Problems149

10.FunctionSpaces153

10.1.BasicInequalitiesandDefinitions153



10.2.Multi-IndexNotation157

10.3.SobolevSpacesWk,p(U)158

Problems159

11.EllipticTheorywithSobolevSpaces161

11.1.Poisson’sEquation161

11.2.LinearSecond-OrderEllipticEquations167

Problems173

12.TravelingWaveSolutionsofPDE175

12.1.Burgers’Equation175

12.2.TheKorteweg-deVriesEquation176

12.3.Fisher’sEquation179

12.4.TheBistableEquation181

Problems186

13.ScalarConservationLaws189

13.1.TheInviscidBurgersEquation189

13.2.ScalarConservationLaws196

13.3.TheLaxEntropyConditionRevisited201

13.4.UndercompressiveShocks204

13.5.The(Viscous)BurgersEquation206

13.6.MultidimensionalConservationLaws208

Problems211

14.SystemsofFirst-OrderHyperbolicPDE215

14.1.LinearSystemsofFirst-OrderPDE215

14.2.SystemsofHyperbolicConservationLaws219

14.3.TheDam-BreakProblemUsingShallowWaterEquations239

14.4.Discussion241

Problems242

15.TheEquationsofFluidMechanics245

15.1.TheNavier-StokesandStokesEquations245

15.2.TheEulerEquations247

Problems250



AppendixA.MultivariableCalculus253

AppendixB.Analysis259

AppendixC.SystemsofOrdinaryDifferentialEquations263

References265

Index269



Preface

Thefieldofpartialdifferentialequations(PDEforshort)hasalonghistorygoing

backseveralhundredyears,beginningwiththedevelopmentofcalculus.Inthis

regard,thefieldisatraditionalareaofmathematics,althoughmorerecentthan

suchclassicalfieldsasnumbertheory,algebra,andgeometry.Asinmanyareas

ofmathematics,thetheoryofPDEhasundergonearadicaltransformationinthe

past hundred years, fueled by the development of powerful analytical tools,

notably, the theory of functional analysis and more specifically of function

spaces.Thedisciplinehasalsobeendrivenbyrapiddevelopmentsinscienceand

engineering, which present new challenges of modeling and simulation and

promotebroaderinvestigationsofpropertiesofPDEmodelsandtheirsolutions.

AsthetheoryandapplicationofPDEhavedeveloped,profoundunanswered

questions and unresolved problems have been identified. Arguably the most

visible is one of the Clay Mathematics Institute Millennium Prize problems1

concerning the Euler and Navier-Stokes systems of PDE that model fluid flow.

The Millennium problem has generated a vast amount of activity around the

world in an attempt to establish well-posedness, regularity and global existence

results, not only for the Navier-Stokes and Euler systems but also for related

systemsofPDEmodelingcomplexfluids(suchasfluidswithmemory,polymeric

fluids, and plasmas). This activity generates a substantial literature, much of it

highly specialized and technical. Meanwhile, mathematicians use analysis to

probenewapplicationsandtodevelopnumericalsimulationalgorithmsthatare

provably accurate and efficient. Such capability is of considerable importance,

given the explosion of experimental and observational data and the spectacular

accelerationofcomputingpower.

OurtextprovidesagatewaytothefieldofPDE.Weintroducethereadertoa

varietyofPDEandrelatedtechniquestogiveasenseofthebreadthanddepthof

thefield.Weassumethatstudentshavebeenexposedtoelementaryideasfrom

ordinarydifferentialequations(ODE)andanalysis;thus,thebookisappropriate

foradvancedundergraduateorbeginninggraduatemathematicsstudents.Forthe

studentpreparingforresearch,weprovideagentleintroductiontosomecurrent

theoretical approaches to PDE. For the applied mathematics student more

interested in specific applications and models, we present tools of applied

mathematicsinthesettingofPDE.Scienceandengineeringstudentswillfinda

rangeoftopicsinthemathematicsofPDE,withexamplesthatprovidephysical

intuition.

Our aim is to familiarize the reader with modern techniques of PDE,



introducing abstract ideas straightforwardly in special cases. For example,

strugglingwiththedetailsandsignificanceofSobolevembeddingtheoremsand

estimates is more easily appreciated after a first introduction to the utility of

specific spaces. Many students who will encounter PDE only in applications to

scienceandengineeringorwhowanttostudyPDEforjustayearwillappreciate

this focused, direct treatment of the subject. Finally, many students who are

interested in PDE have limited experience with analysis and ODE. For these

students, this text provides a means to delve into the analysis of PDE before or

while taking first courses in functional analysis, measure theory, or advanced

ODE.BasicbackgroundonfunctionsandODEisprovidedinAppendicesA–C.

To keep the text focused on the analysis of PDE, we have not attempted to

include an account of numerical methods. The formulation and analysis of

numerical algorithms is now a separate and mature field that includes major

developmentsintreatingnonlinearPDE.However,thetheoreticalunderstanding

gained from this text will provide a solid basis for confronting the issues and

challengesinnumericalsimulationofPDE.

A student who has completed a course organized around this text will be

prepared to study such advanced topics as the theory of elliptic PDE, including

regularity,spectralproperties,therigoroustreatmentofboundaryconditions;the

theoryofparabolicPDE,buildingonthesettingofelliptictheoryandmotivating

theabstractideasinlinearandnonlinearsemigrouptheory;existencetheoryfor

hyperbolicequationsandsystems;andtheanalysisoffullynonlinearPDE.

We hope that you, the reader, find that our text opens up this fascinating,

important, and challenging area of mathematics. It will inform you to a level

where you can appreciate general lectures on PDE research, and it will be a

foundationforfurtherstudyofPDEinwhateverdirectionyouwish.

Wearegratefultoourstudentsandcolleagueswhohavehelpedmakethisbook

possible,notablyDavidG.Schaeffer,DavidUminsky,andMarkHoeferfortheir

candid and insightful suggestions. We are grateful for the support we have

received from the fantastic staff at Princeton University Press, especially Vickie

Kern,whohasbelievedinthisprojectfromthestart.

Rachel Levy thanks her parents Jack and Dodi, husband Sam, and children

TulaandMimi,whohavelovinglyencouragedherwork.

Michael Shearer thanks the many students who provided feedback on the

coursenotesfromwhichthisbookisderived.

1.www.claymath.org/millennium/.



PartialDifferentialEquations



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