Partial differential equations an introduction to theory and applications by michael shearer

PartialDifferentialEquations


PartialDifferentialEquations
AnIntroductiontoTheoryandApplications
MichaelShearer
RachelLevy
PRINCETONUNIVERSITYPRESS
PrincetonandOxford


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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


CHAPTERONE

Introduction
Partial differential equations (PDE) describe physical systems, such as solid and
fluid mechanics, the evolution of populations and disease, and mathematical
physics. The many different kinds of PDE each can exhibit different properties.
For example, the heat equation describes the spreading of heat in a conducting
medium,smoothingthespatialdistributionoftemperatureasitevolvesintime;
it also models the molecular diffusion of a solute in its solvent as the
concentrationvariesinbothspaceandtime.Thewaveequationisattheheartof
thedescriptionoftime-dependentdisplacementsinanelasticmaterial,withwave
solutions that propagate disturbances. It describes the propagation of p-waves
ands-wavesfromtheepicenterofanearthquake,theripplesonthesurfaceofa
pondfromthedropofastone,thevibrationsofaguitarstring,andtheresulting
sound waves. Laplace’s equation lies at the heart of potential theory, with
applicationstoelectrostaticsandfluidflowaswellasotherareasofmathematics,
suchasgeometryandthetheoryofharmonicfunctions.ThemathematicsofPDE
includes the formulation of techniques to find solutions, together with the

development of theoretical tools and results that address the properties of
solutions,suchasexistenceanduniqueness.
This text provides an introduction to a fascinating, intricate, and useful
branchofmathematics.Inadditiontocoveringspecificsolutiontechniquesthat
provideaninsightintohowPDEwork,thetextisagatewaytotheoreticalstudies
ofPDE,involvingthefullpowerofreal,complexandfunctionalanalysis.Often
we will refer to applications to provide further intuition into specific equations
andtheirsolutions,aswellastoshowthemodelingofrealproblemsbyPDE.
ThestudyofPDEtakesmanyforms.Verybroadly,wetaketwoapproachesin
thisbook.Oneapproachistodescribemethodsofconstructingsolutions,leading
to formulas. The second approach is more theoretical, involving aspects of
analysis,suchasthetheoryofdistributionsandthetheoryoffunctionspaces.

1.1.LinearPDE
TointroducePDE,webeginwithfourlinearequations.Theseequationsarebasic
to the study of PDE, and are prototypes of classes of equations, each with
different properties. The primary elementary methods of solution are related to
thetechniqueswedevelopforthesefourequations.
For each of the four equations, we consider an unknown (real-valued)


functionuonanopensetU⊂Rn.Werefertouasthedependentvariable,andx
=(x1,x2,…,xn)∈Uasthevectorofindependentvariables.Apartialdifferential
equation is an equation that involves x, u, and partial derivatives of u. Quite
often, x represents only spatial variables. However, many equations are
evolutionary, meaning that u=u(x,t) depends also on time t and the PDE has
time derivatives. The order of a PDE is defined as the order of the highest
derivativethatappearsintheequation.
TheLinearTransportEquation:
This simple first-order linear PDE describes the motion at constant speed c of a

quantityudependingonasinglespatialvariablexandtimet.Eachsolutionisa
travelingwavethatmoveswiththespeedc.Ifc>0,thewavemovestotheright;
ifc<0,thewavemovesleft.Thesolutionsareallgivenbyaformulau(x,t)=
f(x−ct).Thefunctionf=f(ξ),dependingonasinglevariableξ=x−ct,is
determinedfromsideconditions,suchasboundaryorinitialconditions.
Thenextthreeequationsareprototypicalsecond-orderlinearPDE.
TheHeatEquation:
In this equation, u(x, t) is the temperature in a homogeneous heat-conducting
material,theparameterk>0isconstant,andtheLaplacianΔisdefinedby

in Cartesian coordinates. The heat equation, also known as the diffusion
equation,modelsdiffusioninothercontexts,suchasthediffusionofadyeina
clear liquid. In such cases, u represents the concentration of the diffusing
quantity.
TheWaveEquation:
As the name suggests, the wave equation models wave propagation. The
parameter c is the wave speed. The dependent variable u = u(x, t) is a
displacement, such as the displacement at each point of a guitar string as the
string vibrates, if x ∈ R, or of a drum membrane, in which case x ∈ R2. The
acceleration utt, being a second time derivative, gives the wave equation quite
differentpropertiesfromthoseoftheheatequation.


Laplace’sEquation:
Laplace’s equation models equilibria or steady states in diffusion processes, in
whichu(x,t)isindependentoftimet,1andappearsinmanyothercontexts,such
asthemotionoffluids,andtheequilibriumdistributionofheat.
These three second-order equations arise often in applications, so it is very
usefultounderstandtheirproperties.Moreover,theirstudyturnsouttobeuseful
theoretically as well, since the three equations are prototypes of second-order

linearequations,namely,elliptic,parabolic,andhyperbolicPDE.

1.2.Solutions;InitialandBoundaryConditions
AsolutionofaPDEsuchasanyof(1.1)–(1.4)isareal-valuedfunctionusatisfying
theequation.OftenthismeansthatuisasdifferentiableasthePDErequires,and
the PDE is satisfied at each point of the domain of u. However, it can be
appropriate or even necessary to consider a more general notion of solution, in
whichuisnotrequiredtohaveallthederivativesappearingintheequation,at
leastnotintheusualsenseofcalculus.Wewillconsiderthiskindofweaksolution
later(seeChapter11).
As with ordinary differential equations (ODE), solutions of PDE are not
unique;identifyingauniquesolutionreliesonsideconditions,suchasinitialand
boundary conditions. For example, the heat equation typically comes with an
initialconditionoftheform
inwhichu0:U→Risagivenfunction.
Example1.(Simpleinitialcondition)Thefunctionsu(x,t)=ae−tsinx+be−4t
sin(2x) are solutions of the heat equation ut = uxx for any real numbers a, b.
However,a=3,b=−7wouldbeuniquelydeterminedbytheinitialcondition
u(x,0)=3sinx−7sin(2x).Thenu(x,t)=3e−tsinx−7e−4tsin(2x).
Boundary conditions are specified on the boundary ∂U of the (spatial)
domain.Dirichletboundaryconditionstakethefollowingform,foragivenfunction
f:∂U→R:
Neumannboundaryconditionsspecifythenormalderivativeofuontheboundary:


where ν(x) is the unit outward normal to the boundary at x. These boundary
conditions are called homogeneous if f ≡ 0. Similarly, a linear PDE is called
homogeneousifu=0isasolution.Ifitisnothomogeneous,thentheequationor
boundaryconditioniscalledinhomogeneous.
Equationsandboundaryconditionsthatarelinearandhomogeneoushavethe

propertythatanylinearcombinationu=av+bwofsolutionsv,w,witha,b∈
R, is also a solution. This special property, sometimes called the principle of
superposition,iscrucialtoconstructivemethodsofsolutionforlinearequations.

1.3.NonlinearPDE
We introduce a selection of nonlinear PDE that are significant by virtue of
specificproperties,specialsolutions,ortheirimportanceinapplications.
TheInviscidBurgersEquation:
is an example of a nonlinear first-order equation. Notice that this equation is
nonlinearduetotheuuxterm.Itisrelatedtothelineartransportequation(1.1),
but the wave speed c is now u and depends on the solution. We shall see in
Chapter 3 that this equation and other first-order equations can be solved
systematicallyusingaprocedurecalledthemethodofcharacteristics.However,the
method of characteristics only gets you so far; solutions typically develop a
singularity,inwhichthegraphofuasafunctionofxsteepensinplacesuntilat
some finite time the slope becomes infinite at some x. The solution then
continueswithashockwave.Thesolutionisnotevencontinuousattheshock,
but the solution still makes sense, because the PDE expresses a conservationlaw
andtheshockpreservesconservation.
For higher-order nonlinear equations, there are no methods of solution that
work in as much generality as the method of characteristics for first-order
equations.Hereisasampleofhigher-ordernonlinearequationswithinteresting
andaccessiblesolutions.
Fisher’sEquation:
withf(u)=u(1−u). This equation is a model for population dynamics when
the spatial distribution of the population is taken into account. Notice the
resemblance to the heat equation; also note that the ODE u′(t) = f(u(t)) is the
logisticequation,describingpopulationgrowthlimitedbyamaximumpopulation
normalized to u = 1. In Chapter12, we shall construct traveling waves, special

solutions in which the population distribution moves with a constant speed in
onedirection.Recallthatallsolutionsofthelineartransportequation(1.1)are
traveling waves, but they all have the same speed c. For Fisher’s equation, we
havetodeterminethespeedsoftravelingwavesaspartoftheproblem,andthe
travelingwavesarespecialsolutions,notthegeneralsolution.
ThePorousMediumEquation:
Inthisequation,m>0isconstant.Theporousmediumequationmodelsflowin
porousrockorcompactedsoil.Thevariableu(x,t)≥0measuresthedensityofa
compressiblegasinagivenlocationxattimet.Thevalueofmdependsonthe
equation of state relating pressure in the gas to its density. For m = 1, we
recovertheheatequation,butform≠1,theequationisnonlinear.Infact,m≥
2forgasflow.
TheKorteweg-deVries(KdV)Equation:
Thisthird-orderequationisamodelforwaterwavesinwhichtheheightofthe
wave is u(x, t). The KdV equation has particularly interesting traveling wave
solutions called solitarywaves, in which the height is symmetric about a single
crest.Theequationisamodelinthesensethatitreliesonanapproximationof
the equations of fluid mechanics in which the length of the wave is large
comparedtothedepthofthewater.
Burgers’Equation:
The parameter ν > 0 represents viscosity, hence the name inviscid Burgers
equationforthefirst-orderequation(1.6)havingν = 0. Burgers’ equation is a
combination of the heat equation with a nonlinear term that convects the
solutioninawaytypicaloffluidflow.(SeetheNavier-Stokessystemlaterinthis
list.)Thisimportantequationcanbereducedtotheheatequationwithaclever
change of dependent variable, called the Cole-Hopf transformation (see Chapter
13,Section13.5).
Finally,wementiontwosystemsofnonlinearPDE.
TheShallowWaterEquations:

in which g > 0 is the gravitational acceleration. The dependent variables h, v


representtheheightandvelocity,respectively,ofashallowlayerofwater.The
variablexisthehorizontalspatialvariable,alongaflatbottom,anditisassumed
that there is no dependence or motion in the orthogonal horizontal direction.
Moreover,thevelocityvistakentobeindependentofdepth.
TheNavier-StokesEquations:

describe the velocity u ∈ R3 and pressure p in the flow of an incompressible

viscous fluid. In this system of four equations, the parameter ν > 0 is the
viscosity,thefirstthreeequations(foru)representconservationofmomentum,
andthefinalequationisaconstraintthatexpressestheincompressibilityofthe
fluid. In an incompressible fluid, local volumes are unchanged in time as they
follow the flow. Apart from special types of flow (such as in a stratified fluid),
incompressibilityalsomeansthatthedensityisconstant(andisincorporatedinto
ν,thekinematicviscosity).
Interestingly,themomentumequation,regardedasanevolutionequationfor
u,resemblesBurgers’equationinstructure.Thepressurepdoesnothaveitsown
evolutionequation;itservesmerelytomaintainincompressibility.Inthelimitν
→0,werecovertheincompressibleEulerequationsforaninviscidfluid.Thisisa
singularlimitinthesensethattheorderofthemomentumequationisreduced.It
isalsoasingularlimitforBurgers’equation.

1.4.BeginningExampleswithExplicitWave-likeSolutions
ThelinearandnonlinearfirstorderequationsdescribedinSections1.1and1.3
nicely illustrate mathematical properties and representation of wave-like
solutions. We discuss these equations and their solutions as a starting point for

moregeneralconsiderations.
1.4.1.TheLinearTransportEquation
Solutionsofthelineartransportequation,
wherec∈Risaconstant(thewavespeed),aretravelingwavesu(x,t)=f(x−
ct). We can determine a unique solution by specifying the function f : R → R
fromaninitialcondition


Figure1.1.Lineartransportequation:travelingwavesolution.(a)t=0;(b)t>
0.
inwhichu0:R→Risagivenfunction.Thentheuniquesolutionoftheinitial
valueproblem (1.8), (1.9) is the traveling wave u(x, t) = u0(x − ct). A typical
travelingwaveisshowninFigure1.1.
Insteadofinitialconditions,wecanalsospecifyaboundaryconditionforthis
PDE.Hereisanexampleofhowthiswouldlook,forfunctionsϕ,ψgivenonthe
interval[0,∞):

Thesolutionuof(1.8),(1.10)willbeafunctiondefinedonthefirstquadrantQ1
={(x,t):x≥0,t≥0}inthex-tplane.ThegeneralsolutionofthePDEisu(x,
t)=f(x − ct); the initial condition specifies f(y) for y > 0, and the boundary
conditiongivesf(y)fory<0.Bothareneededtodeterminethesolutionu(x,t)
onQ1.
1.4.2.TheInviscidBurgersEquation
Thisequation,
haswavespeeduthatdependsonthesolution,incontrasttothelineartransport
equation(1.8)inwhichthewavespeedcisconstant.Ifweusethewavespeedto
track the solution, we can sketch its evolution. In Figure1.2 we show how an
initial condition (1.9) evolves for small t > 0. Points nearer the crest travel
faster,sinceuislargerthere,sothefrontofthewavetendstosteepen,whilethe
backspreadsout.NoticehowFigure1.2differsfromFigure1.1.Thesolutionu=

u(x,t)canbespecifiedimplicitlyinanequationwithoutderivatives:


Figure1.2.InviscidBurgersequation:nonlinearwavepropagation.(a)t=0;(b)
t>0.
Eventually,thegraphbecomesinfinitelysteep,andtheimplicitsolutionin(1.12)
isnolongervalid.Thesolutioniscontinuedtolargertimebyincludingashock
wave,definedinChapter13.

PROBLEMS
1.Showthatthetravelingwaveu(x,t)=f(x−3t)satisfiesthelineartransport
equationut+3ux=0foranydifferentiablefunctionf:R→R.
2.Findanequationrelatingtheparametersk,m,nsothatthefunctionu(x,t)=
emtsin(nx)satisfiestheheatequationut=kuxx.
3.Findanequationrelatingtheparametersc,m,nsothatthefunctionu(x,t)=
sin(mt)sin(nx)satisfiesthewaveequationutt=c2uxx.
4.Findallfunctionsa,b,c:R→Rsuchthatu(x,t)=a(t)e2x+b(t)ex+c(t)
satisfiestheheatequationut=uxxforallx,t.
5. For m > 1, define the conductivity k = k(u) so that the porous medium
equation(1.7)canbewrittenasthe(quasilinear)heatequation
6.Solvetheinitialvalueproblem

7.Solvetheinitialboundaryvalueproblem

ExplainwhythereisnosolutionifthePDEischangedtout−4ux=0.


8. Consider the linear transport equation (1.8) with initial and boundary
conditions(1.10).
(a)Supposethedataϕ,ψaredifferentiablefunctions.Showthatthefunction

u:Q1→Rgivenby

satisfies the PDE away from the line x = ct, the boundary condition, and
initial condition. To see where (1.13) comes from, start from the general
solutionu(x,t)=f(x−ct)ofthePDEandsubstituteintothesideconditions
(1.10).
(b)Insolution(1.13),thelinex=ct,whichemergesfromtheoriginx=t=
0, separates the quadrant Q1 into two regions. On the line, the solution has
one-sidedlimitsgivenbyϕ,ψ.Consequently,thesolutionwillingeneralhave
singularitiesontheline.
(i) Find conditions on the data ϕ, ψ so that the solution is continuous
acrossthelinex=ct.
(ii)Findconditionsonthedataϕ,ψsothatthesolutionisdifferentiable
acrossthelinex=ct.
9. Let f : R → R be differentiable. Verify that if u(x, t) is differentiable and
satisfies(1.12),thatis,u=f(x−ut),thenu(x,t)isasolutionoftheinitialvalue
problem
10.Letu0(x)=1−x2if−1≤x≤1,andu0(x)=0otherwise.
(a) Use (1.12) to find a formula for the solution u = u(x, t) of the inviscid
Burgersequation(1.11),(1.9)with−1.
(b)Verifythatu(1,t)=0,

.

(c) Differentiate your formula to find ux(1−,t), and deduce that ux(1−,t) →
−∞as
.
Note:ux(x,t) is discontinuous at x = ±1; the notation u(1−, t) means the
one-sidedlimit:

.Similarly,
means,
,with
.
1.However,therearetime-dependentsolutions,forexampleu(x,t)linearinxorindependentofx.


CHAPTERTWO

Beginnings
In the previous chapter we constructed solutions for example equations.
However, much of the study of PDE is theoretical, revolving around issues of
existenceanduniquenessofsolutions,andpropertiesofsolutionsderivedwithout
writingformulasforthesolutions.Ofcourse,existenceanduniquenessissuesare
resolvedifitispossibletoconstructallsolutionsofagivenPDE,butcommonly
thisconstructiveapproachisnotavailable,andmoreabstractmethodsofanalysis
arerequired.Inthischapterweoutlinetheoreticalconsiderationsthatwillcome
upfromtimetotime,giveasomewhatgeneralclassificationofsingleequations,
and then give a flavor of theoretical approaches by presenting the CauchyKovalevskayatheoremanddiscussingsomeofitsramifications.Finally,weshow
how PDE can be derived from balance laws (otherwise known as conservation
laws) that come from fundamental considerations underlying the modeling of
mostapplications.

2.1.FourFundamentalIssuesinPDETheory
Generally, the theoretical study of PDE focuses on four basic issues, three of
whicharelumpedtogetheraswell-posednessinthesenseofHadamard.1
1.Existence:IsthereasolutionofthePDEsatisfyingaspecificsetofboundary
andinitialconditions?
2.Uniqueness:Isthereonlyonesolutionforaspecificsetofboundaryandinitial
conditions?

3.Continuousdependenceondata:Dosmallchangesininitialconditions,
boundaryconditions,andparameterscreateonlysmallchangesinthe
solution?Wemightsaythesolutionisrobusttochangesinthedata.
Sometimes,thispropertyiscalledstructuralstability,ormoreloosely,stability.
Thefourthpropertyisgenerallyseparatedfromconsiderationsofwell-posedness:
4.Regularity:Howmanyderivativesdoesthesolutionhave?Wesometimesrefer
tothispropertyasthesmoothnessofthesolution.
Well-posedness is a desirable property if the goal is to model a repeatable
experiment, for example. Of the four properties, one could argue that the most
importantpropertyisexistence.Afterall,whatuseisaPDEmodelifitdoesnot
have a solution? In the theory of ODE, showing the existence of solutions is
generally straightforward, at least locally, based on the classical existence and


uniqueness theorem for initial value problems. In the previous chapter we
establishedexistencebyconstructingsolutions.However,ingeneralthetheoryof
existenceofsolutionsforPDEisacomplexandhighlytechnicalsubject.
Existence.Theapproachofthisbookistostudyexistenceissuesonlyforclasses
ofequations(andclassesofsolutions)forwhichthetheoryiselementary,suchas
classical (i.e., continuously differentiable) solutions of first-order equations. For
second-order equations, we begin by choosing problems for which we can
construct explicit solutions, thus avoiding the technicalities of proving general
existencetheorems.Towardtheendofthebook(seeChaps.9–11),weintroduce
some of the theoretical underpinnings of more general theories of PDE, such as
thetheoryofdistributions,theuseofSobolevspaces,andmaximumprinciples.
Uniqueness. Uniqueness is often the easiest property to establish. Moreover, it
doesnotrequiretheexistenceofsolutions,aswecanstate:“Thereexistsatmost
onesolution.”
Continuous dependence. Continuous dependence can be established using
techniques from analysis that estimate the closeness of distinct solutions with

differentdata,intermsoftheclosenessofthedata.Closenessofcourseinvolves
defining a suitable notion of distance—a metric—on both the space in which
solutions reside and on the space of data. These notions will be formally
introducedasneeded.
Regularity. Regularity is generally the hardest property to characterize,
requiring the most delicate analysis. In this text we make observations about
regularity from explicit solutions; regularity more generally and theoretically
involvesmoretechnicalmachinery.

2.2.ClassificationofSecond-OrderPDE
When studying ODE, it is convenient to be able to distinguish among different
kindsofequationsbasedonsuchcriteriaaslinearvs.nonlinearandseparablevs.
nonseparable. For PDE, there are also multiple ways to distinguish among
equations, some similar to the criteria for ODE. In the next chapter we discuss
first-orderPDEindetail,showingthatthetheoryislinkedcloselytosystemsof
first-orderODE.
For second-order equations, there are distinct families of equations,
distinguished by typical properties of their solutions. We identify the class of
hyperbolic equations, with wave-like solutions, and elliptic equations,
representing steady-state or equilibrium solutions. Between these two general
classes are the parabolic equations, which, like hyperbolic equations, have a


time-like independent variable but also have properties akin to those of elliptic
equations. The heat equation, the wave equation, and Laplace’s equation are
second-order linear constant-coefficient prototypes of parabolic, hyperbolic, and
elliptic PDE, respectively. Although this chapter is primarily about linear
equationsintwovariables,weincludesomeremarksaboutequationswithmore
independentvariablesandnonlinearequations.
2.2.1.ConstantCoefficients

Toexplainhowthetermshyperbolic,elliptic,andparaboliccometobeassociated
withPDE,itissimplesttoconsiderasecond-orderequationoftheform
wherethecoefficientsa,b,carerealnumbers,andtheright-handsidef=f(x,y,
u,ux,uy)isagivenfunctioncontaininganylower-orderderivativesofu.Thetype
oftheequationisdeterminedbythenatureofthequadraticformobtainedfrom
theleft-handsideof(2.1)byreplacingeachpartialderivativebyarealvariable.
More formally, we define the principal part of the PDE as the left-hand side of
(2.1).Thenthecorrespondingdifferentialoperatorwithprincipalindicatedbythe
superscript(p)is
Associated with this differential operator is the quadratic form, known as the
principalsymbol,
inwhichξ=(ξ1,ξ2)∈R2.Theconnectionbetweenprincipalpartandprincipal
symbolistheobservation
This conversion from differential operators ∂x,∂y to multiplication by iξ1, iξ2 is
typical of integral transforms; in this case, the connection is to Fourier
transforms.Thevector(ξ1,ξ2)istheFouriertransformvariable,orwavenumber.
FouriertransformsandtheirimportancefortheanalysisofPDEarediscussedin
Chapter7.
Thequadraticform(2.2)isassociatedwitheitherahyperbola(ifb2>ac),an
ellipse (if b2 < ac), or is degenerate (if b2 = ac). Correspondingly, we say the
PDE (2.1) is hyperbolic if b2 > ac, elliptic if b2 < ac, and parabolic if b2 = ac,
providedtheequationissecondorder(i.e.,notallofa,b,carezero).
Example1.(Classification)ThepartialdifferentialoperatorL=∂2x+α∂2y,is


ellipticforα>0,hyperbolicforα<0,andparabolicforα=0.
2.2.2.MoreGeneralSecond-OrderEquations
A similar classification applies to second-order equations in any number of
variables.Asusual,writex=(x1,x2,…,xn)∈Rn.Considertheequation

wheref=f(x,u,ux1,…,uxn).Weassumetherealcoefficientsaijintheprincipal
partL(p)u(givenbytheleft-handside)areconstantandsymmetricini,j:aij=aji.
(If they were not symmetric, we could rearrange the PDE using the equality of
mixedpartialderivativestoachievesymmetry.)Theprincipalsymbolisthen

The type of the PDE depends on the nature of this quadratic expression, which
wecanwriteinmatrixform:
where A = (aij) is a real symmetric n × n matrix. If we change independent
variableswithaninvertiblelineartransformationB,
then the chain rule changes the PDE (2.3). It is instructive (see Problem 2) to
work out that the principal symbol now has coefficient matrix BABT. If B is an
orthogonal matrix, then B−1 = BT, so that the linear change of independent
variablescorrespondstoasimilaritytransformationofA.Nowlet’schooseBto
diagonalizeA,sothatBABThastheneigenvaluesofAonthediagonalandzeroes
elsewhere. This is achieved by letting the columns of B be the orthonormal
eigenvectorsofA. The effect on the PDE is to convert the principal part into a
linearcombinationofpuresecond-orderderivatives,inwhichthecoefficientsare
theeigenvaluesofA.
WesaythePDEisellipticiftheeigenvaluesofAareallnonzero,andallhave
thesamesign.ThePDEiscalledhyperbolicifalleigenvaluesarenonzero,andall
butoneofthemhavethesamesign.(Thereisthethirdpossibilitythat,forn≥
4, all but k eigenvalues, with 2 ≤ k ≤ n/2, have the same sign. This case is
called ultrahyperbolic, but it does not occur much, so we ignore it.) Finally, if
there is at least one zero eigenvalue, then we could consider the PDE to be
parabolic. In practice, parabolic equations occur most commonly as time-


dependent PDE like the heat equation, with a single zero eigenvalue. Such
parabolicequationstypicallyhavetheform
where u = u(x, t), L is a linear elliptic operator with respect to the spatial

variables,andf=f(x,t,u,ux1,…,uxn).Inthisequation,onlyoneeigenvalueof
thecoefficientmatrixAiszero.
For each type of linear second-order PDE, we can find a change of
independentvariablestotransformtheequationintoacanonicalform,inwhich
the corresponding matrix A is diagonal, so that only pure second-order
derivativesoccur(i.e.,nocrossderivatives).Infact,thechangeofvariablescan
be done in general by observing how a linear change of independent variables
correspondstoasimilaritytransformationofA.Thenwecanreversetheprocess
tofindtheappropriatechangeofvariablesfromadiagonalizationofA.
Letx ∈ Rn be the independent variable, and suppose we introduce a linear
changeofvariablestoy,throughtheorthogonalmatrixBdefinedabove,sothat
BABTisdiagonal:
In coordinates, this reads
. If u = u(x), we define w(y) = u(Cy),
whereC=B−1.Thenacarefulcalculationgives

whereλ1,…,λnaretheeigenvaluesofA.
Example2.(SamplePDEoperators)Let’sadoptthenotation∂jinterchangeably
with∂/∂xj.HerewedisplayaPDEoperator,thecorrespondingmatrixA,andthe
typeoftheoperator:
1.
2.

;elliptic.
;x1=t,x2=x,x3=y;

;hyperbolic.

Noticethatforahyperbolicequation,theoneeigenvaluewithadifferentsign
suggestsatime-likedirection(associatedwiththecorrespondingeigenvalue).

AfterdiagonalizingA,wecanscaleeachindependentvariablesothatinthe
newvariables,wehave