Applied Partial Differential Equations

JOHN OCKENDON Oxford Centrefor Industrial and Applied Mathematics

University of Oxford

SAM HOWISON Oxford Centrefor Industrial and Applied Mathematics

University of Oxford

ANDREW LACEY Department of Mathematics

Heriot- Watt University

and

ALEXANDER MOVCHAN Department of Mathematical Sciences

Liverpool University

OXTORD UNIVERSITY PRESS

Contents

Introduction

First-order scalar quasilinear equations 1.1 1.2 1.3

1.4 1.5 1.6

*1.7 *1.8

1.9

Introduction Cauchy data Characteristics 1.3.1 Linear and semilinear equations Domain of definition and blow-up Quasilinear equations Solutions with discontinuities Weak solutions More independent variables Postscript

First-order quasilinear Systems 2.1 2.2 2.3 2.4

*2.5

*2.6

Motivation and modeis Cauchy data and characteristics The Cauchy-Kowalevski theorem Hyperbolicity 2.4.1 Two-by-two Systems 2.4.2 Systems of dimension n 2.4.3 Examples Weak solutions and shock waves 2.5.1 Causality 2.5.2 Viscosity and entropy 2.5.3 Other discontinuities Systems with more than two independent variables

Introduction to second-order scalar equations 3.1 3.2 3.3 3.4

3.5

Preamble The Cauchy problem for semilinear equations Characteristics Canonical forms for semilinear equations 3.4.1 Hyperbolic equations 3.4.2 Elliptic equations 3.4.3 Parabolic equations Some general remarks

1

6 6 8 9

11 13 15 19 22 25 28

35 35 41 45 48 49 50 52 55 56 60 61 63

75 75 77 79 81 81 83 85 86

VU1 Contents

4 Hyperbolic equations 92 4.1 Introduction 92 4.2 The Cauchy problem 93

4.2.1 An ad hoc approach to Riemann functions 93 4.2.2 The rationale for Riemann functions 95 4.2.3 Implications of the Riemann function representation 99

4.3 Non-Cauchy data for the wave equation 101 *4.3.1 Strongly discontinuous boundary data 104

4.4 Transforms and eigenfunction expansions 105 4.5 Applications to wave equations 110

4.5.1 One space dimension 110 4.5.2 Circular and spherical symmetry 112

*4.5.3 The telegraph equation 114 *4.5.4 Waves in periodic media 115 *4.5.5 General remarks 116

4.6 More than two independent variables 117 4.6.1 The method of descent and Huygens' principle 117 4.6.2 Hyperbolicity and time-likeness 121

*4.7 Higher-order Systems 123 4.7.1 Linear elasticity 123 4.7.2 Maxwell's equations of electromagnetism 126

4.8 Nonlinearity 129 4.8.1 Simple waves 130 4.8.2 Hodograph methods 132 4.8.3 Liouville's equation 134

*4.8.4 Another method 135

5 Elliptic equations 144 5.1 Models 144

5.1.1 Gravitation 144 5.1.2 Electromagnetism 145 5.1.3 Heattransfer 146 5.1.4 Mechanics 148 5.1.5 Acoustics 153 5.1.6 Aerofoil theory and fracture 154

5.2 Well-posed boundary data 156 5.2.1 The Laplace and Poisson equations 156 5.2.2 More general elliptic equations 159

5.3 The maximum principle 160 5.4 Variational principles 161 5.5 Green's functions 162

5.5.1 The classical formulation 162 5.5.2 Generalised function formulation 164

Contents ix

5.6 Explicit representations of Green's functions 167 5.6.1 Laplace's equation and Poisson's equation 167 5.6.2 Helmholtz' equation 173 5.6.3 The modified Helmholtz equation 174

*5.7 Transforms 175 5.7.1 Eigenvalues and eigenfunctions 175 5.7.2 Green's functions and transforms 176

5.8 Transform Solutions of elliptic problems 178 5.8.1 Laplace's equation with cylindrical symmetry: Han-

kel transforms 179 5.8.2 The Meilin transform 181

*5.8.3 Helmholtz' equation 183 *5.8.4 Higher-order problems 186

5.9 Complex variable methods 187 5.9.1 Conformal maps 188

*5.9.2 Riemann-Hilbert problems 189 *5.9.3 Mixed boundary value problems 195 *5.9.4 The Wiener-Hopf method 197 *5.9.5 Singularities and index 199

*5.10 Localised boundary data 201 5.11 Nonlinear problems 202

5.11.1 Nonlinear modeis 203 5.11.2 Existence and uniqueness 204 5.11.3 Parameter dependence and singular behaviour 205

5.12 Liouville's equation again 211 5.13 Postscript: V2 o r - A ? 212

Parabolic equations 226 6.1 Linear modeis ofdiffusion 226

6.1.1 Heat and mass transfer 226 6.1.2 Probability and finance 228 6.1.3 Electromagnetism 230 6.1.4 General remarks 230

6.2 Initial and boundary conditions 230 6.3 Maximum principles and well-posedness 232

*6.3.1 The strong maximum principle 233 6.4 The heat equation 234

6.4.1 Green's functions: general remarks 234 6.4.2 Explicit representations 236 6.4.3 Boundary value problems 239

* 6.4.4 Convection-diffusion problems 245

X Contents

6.5 Similarity Solutions and groups 247 6.5.1 Ordinary differential equations 250 6.5.2 Partial differential equations 250

*6.5.3 General remarks 254 6.6 Nonlinear equations 256

6.6.1 Models 256 6.6.2 Theoretical remarks 260 6.6.3 Similarity Solutions and travelling waves 260 6.6.4 Comparison methods and the maximum principle 266

*6.6.5 Blow-up 269 * 6.7 Higher-order equations and Systems 272

6.7.1 Higher-order scalar problems 272 6.7.2 Higher-order Systems 274

Free boundary problems 290 7.1 Introduction and modeis 290

7.1.1 Stefan and related problems 291 7.1.2 Other free boundary problems in diffusion 295 7.1.3 Some other problems from mechanics 299

7.2 Stability and well-posedness 303 7.2.1 Surface gravity waves 305 7.2.2 Vortex sheets 306 7.2.3 Hele-Shaw flow 308 7.2.4 Shock waves 309

7.3 Classical Solutions 311 7.3.1 Comparison methods 311 7.3.2 Energy methods and conserved quantities 312 7.3.3 Green's functions and integral equations 313

*7.4 Weak and variational methods 314 7.4.1 Variational methods 315 7.4.2 The enthalpy method 320

7.5 Explicit Solutions 323 7.5.1 Similarity Solutions 323

*7.5.2 Complex variable methods 326 * 7.6 Regularisation 330 *7.7 Postscript 331

Non-quasilinear equations 342 8.1 Introduction 342 8.2 Scalar first-order equations 343

8.2.1 Two independent variables 343 8.2.2 More independent variables 348 8.2.3 The eikonal equation 349

* 8.2.4 Eigenvalue problems 356 8.2.5 Dispersion 358

Contents XI

*8.3 Hamilton-Jacobi equations and quantum mechanics 359 *8.4 Higher-order equations 362

*9 Miscellaneous topics 373 9.1 Introduction 373 9.2 Linear Systems revisited 375

9.2.1 Linear Systems: Green's functions 376 9.2.2 Linear elasticity 378 9.2.3 Linear inviscid hydrodynamics 380 9.2.4 Wave propagation and radiation conditions 383

9.3 Quasilinear Systems with one real characteristic 384 9.3.1 Heat conduction with ohmic heating. 384 9.3.2 Space Charge 385 9.3.3 Fluid dynamics: the Navier-Stokes equations 386 9.3.4 Inviscid flow; the Euler equations 386 9.3.5 Viscous flow 389

9.4 Interaction between media 391 9.4.1 Fluid/solid acoustic interactions 391 9.4.2 Fluid/fluid gravity wave interaction 392

9.5 Gauges and invariance 392 9.6 Solitons 394

Conclusion 411

References 413

Index 417