Elliptic–Hyperbolic Partial Differential Equations: A Mini-Course in Geometric and Quasilinear Methods

134 Pages·2015·2.12 MB·English

by  Thomas H. Otway (auth.)

SPRINGER BRIEFS IN MATHEMATICS
Thomas H. Otway
Elliptic–Hyperbolic
Partial Differential
Equations
A Mini-Course in
Geometric and
Quasilinear Methods
123
SpringerBriefs in Mathematics
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Thomas H. Otway
–
Elliptic Hyperbolic Partial
Differential Equations
A Mini-Course in Geometric and Quasilinear
Methods
123
ThomasH.Otway
Department ofMathematical Sciences
Yeshiva University
NewYork,NY
USA
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-19760-9 ISBN978-3-319-19761-6 (eBook)
DOI 10.1007/978-3-319-19761-6
LibraryofCongressControlNumber:2015941130
MathematicalSubjectClassification:35M10,35M12,35M30,35M32
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Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Nature of the Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 What Has Been Left Out?. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Organization of the Course. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Overview of Elliptic–Hyperbolic PDE . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Remarks on Equation Type. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 A Zoo of Elliptic–Hyperbolic Equations. . . . . . . . . . . . . . . . . . 12
2.3 A Simple Elliptic–Hyperbolic System. . . . . . . . . . . . . . . . . . . . 14
2.3.1 Boussinesq Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 A Quasilinear Elliptic–Hyperbolic Equation Having
Multiple Sonic Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Surfaces of Prescribed Gauss Curvature . . . . . . . . . . . . . 19
2.5 Local Canonical Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.1 Why Do Equations of Keldysh Type
Have Weaker Regularity?. . . . . . . . . . . . . . . . . . . . . . . 22
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Hodograph and Partial Hodograph Methods . . . . . . . . . . . . . . . . . 31
3.1 The Hodograph and Legendre Transformations . . . . . . . . . . . . . 31
3.2 The Steady Transonic Small-Disturbance Equation. . . . . . . . . . . 33
3.2.1 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The Busemann Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 The Hodograph Image of the Compressible
Flow Equations Acquires the Geometry
of the Projective Disc. . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.2 Transformation to the Lavrent’ev–Bitsadze Equation . . . . 40
3.3.3 Quasilinear Equations in the Hodograph Plane . . . . . . . . 41
v
vi Contents
3.4 An Alternative: Explicit Solutions for a Class
of Quasilinear Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Free Boundary Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 A Partial Hodograph Method for a Nonlinear
Lavrent’ev–Bitsadze Equation. . . . . . . . . . . . . . . . . . . . . . . . . 48
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 A Quasilinear Tricomi Problem. . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Ingredients of the Proof . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.2 Remarks on Generalizations . . . . . . . . . . . . . . . . . . . . . 59
4.1.3 Remarks on Lavrent’ev–Bitsadze Equations . . . . . . . . . . 62
4.2 Examples of Curvature in the Sonic Line . . . . . . . . . . . . . . . . . 63
4.2.1 A Physical Model Having a Parabolic Sonic
Line: Plasma Heating. . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 A Geometric Model Having a Parabolic
Sonic Line: Isometric Embedding . . . . . . . . . . . . . . . . . 65
4.3 A Mixed Dirichlet–Neumann Problem . . . . . . . . . . . . . . . . . . . 66
4.3.1 Function-Space Methods. . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.2 Weak Solutions in Weighted Function Spaces. . . . . . . . . 67
4.3.3 The Existence of Solutions. . . . . . . . . . . . . . . . . . . . . . 68
4.3.4 A Variational Formulation . . . . . . . . . . . . . . . . . . . . . . 73
4.4 The Chaplygin Gas Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.1 Geometric Interpretation. . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.2 Physical Interpretation, and Elliptic–Hyperbolic
Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.3 The Existence of Solutions. . . . . . . . . . . . . . . . . . . . . . 77
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Bäcklund Transformations and Hodge-Theoretic Methods . . . . . . . 81
5.1 Auto-Bäcklund Transformations. . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 An Application to Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 How to Produce Bäcklund Transformations
by Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 A Hierarchy of Variational Problems . . . . . . . . . . . . . . . . . . . . 87
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6 Natural Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1 Introduction to the Isometric Embedding Problem . . . . . . . . . . . 93
6.2 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.1 An Informal Discussion of the Hard Implicit
Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.2 Immersion Versus Embedding. . . . . . . . . . . . . . . . . . . . 103
Contents vii
6.3 Quasilinearization of the Isometric Embedding
Problem I: Changes of Variable. . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Quasilinearization of the Isometric Embedding
Problem II: A Fluid Dynamics Analogy . . . . . . . . . . . . . . . . . . 105
6.5 Energy Condensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5.1 Thin Shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6 Natural Focusing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Appendix A: Informal Review of Differential Forms. . . . . . . . . . . . . . 121
Chapter 1
Introduction
Abstract Thepurposeandstructureofthecoursearediscussedinthisintroductory
chapter. Elliptic–hyperbolic equations are defined, and some examples given. The
expectedbackgroundofthereaderisexplained,asistherelationofthistexttothe
setofICMSworkshoplecturesthatledtoit.Abriefdiscussionisgivenoftopicsthat
hadtobeexcludedfromthecourse.Theorganizationoftopicsisoutlined.
Keywords Elliptic–hyperbolicequation
The main concern of this treatment is to be brief. When encountering a new field,
readerstendtobelessinterestedinlearningallthedetailsoftheproofsthaninfinding
out,initially,whatthemaincontributionsoftheresearchfieldarelikelytobe.This
shortsetofnotes,expandingamini-courseinanICMS workshop,isintendedasa
roadmaptotheemergingfieldofelliptic–hyperbolicgeometricanalysis.
1.1 NatureoftheCourse
Elliptic–hyperbolicequationsarepartialdifferentialequations,definedonadomain
Ω =Ω+∪Ω−∪Ω ,whichareofelliptictypeonthesubdomainΩ+,ofhyperbolic
0
typeonthedisjointsubdomainΩ−,andofparabolictypeonasmoothsubmanifold
Ω whichisasharedboundarybetweenthesubdomainsΩ+andΩ−butwhichlies
0
inneither.Asimpleexampleisanequationhavingtheform
K (x,y)u +u =0, (1.1)
xx yy
where the type-change function K (x,y) is positive in Ω+, negative in Ω−, and
identicallyzeroonthecurveΩ .NoticethatifK (x,y)=1,thenEq.(1.1)reduces
0
totheLaplaceequation,whichisofelliptictype,whereasifK (x,y) = −1,then
(1.1)reducestothewaveequation,whichisofhyperbolictype.Thusanexampleof
atype-changefunctionis[27]
©TheAuthor(s)2015 1
T.H.Otway,Elliptic–HyperbolicPartialDifferentialEquations,
SpringerBriefsinMathematics,DOI10.1007/978-3-319-19761-6_1
2 1 Introduction
Fig.1.1 Theregionsof
ellipticityΩ+,parabolicity
Ω0,andhyperbolicityΩ−
fortheequation
sgn[y]uxx+uyy =0
K (x,y)=sgn[y]; (1.2)
seeFig.1.1.
Elliptic–hyperbolic equations describe, for example, the dynamics of plasmas
subjectedtosmall-amplitudeelectromagneticwaveshavingsufficientlylargephase
velocities in comparison to thermal velocities in the plasma; the behavior of light
near a caustic; extremal surfaces in the space of special relativity; the formation
of rapids; transonic and multiphase fluid flow; the dynamics of certain models for
elasticstructures;theshapeofindustrialsurfacessuchaswindshieldsandairfoils;
pathologiesoftrafficflow;andharmonicfieldsinextendedprojectivespace.They
alsoariseinmodelsfortheearlyuniverse,forcosmicacceleration,andforpossible
violationofcausalityintheinteriorsofcertaincompactstars.Withinthepast25years,
theyhavebecomecentraltotheisometricembeddingofRiemannianmanifoldsand
theprescriptionofGausscurvatureforsurfaces,topicsinpuremathematicswhich
themselveshaveimportantapplications.Referencestotheseandotherapplications
ofelliptic–hyperbolicequationsaregiveninSect.2.2.
Nevertheless,theexpositorymathematicalliteratureonequationsofthiskindis
unaccountablysparse.Thereareseveralmonographswhicharemanydecadesold—
see, e.g., [2, 45], or [49]. There is a large expository literature on the very special
case of elliptic–hyperbolic equations which arise in transonic fluid dynamics, and
in particular, the theory of shock waves—see, e.g., [4, 5, 16, 47, 53], and a host
of other works cited in those references and in [3]. And there are a number of
classictextswhicharebothmanydecadesoldandconsideronlythetransonicspecial
case—see, e.g., [1] and [14]. Han and Hong produced an influential review of the
isometricembeddingproblem[21];but,partlyasaresultofthatreviewandofthe
contributionsofitsauthors,therehasbeenmuchsignificantworkontheisometric
embedding problem since the publication of [21] in 2006. Finally, leaders in the
fieldhaveoccasionallycontributedbriefexpositorynotesontheirwork,c.f.[41,44,
46];these,whileofgreatuse,arebynomeansequivalenttoaformaltreatmentina
monograph.