Partial Differential Equations in Action - From Modelling to Theory

Preface
Contents
Chapter 1 Introduction
1.1 Mathematical Modelling
1.2 Partial Differential Equations
1.3 Well Posed Problems
1.4 Basic Notations and Facts
1.5 Fourier Series
1.6 The Inverse and the Implicit Function Theorems
1.7 Smooth and Lipschitz Domains
1.8 Integration by Parts Formulas
Chapter 2 Diffusion
2.1 The Diffusion Equation
2.1.1 Introduction
2.1.2 The conduction of heat
2.1.3 Initial boundary value problems (n = 1)
2.1.4 A solution by separation of variables
2.1.5 Problems in dimension n > 1
2.2 Uniqueness and Maximum Principles
2.2.1 Integral method
2.2.2 Maximum principles
2.3 The Fundamental Solution
2.3.1 Invariant transformations
2.3.2 The fundamental solution (n - 1)
2.3.3 The Dirac distribution
2.3.4 The fundamental solution (n > 1)
2.4 Symmetric Random Walk (n = 1)
2.4.1 Preliminary computations
2.4.2 From random walk to Brownian motion
2.4.3 A glance at Brownian motion
2.5 Diffusion, Drift and Reaction
2.5.1 Random walk with drift. Preliminary calculations
2.5.2 The limit equation
2.5.3 Pollution in a channel
2.5.4 Random walk with drift and reaction
2.5.5 Critical dimension in a simple population dynamics
2.6 Multidimensional Random Walk
2.6.1 The symmetric case
2.6.2 Walks with drift and reaction
2.7 An Example of Reaction–Diffusion in Dimension n = 3
2.8 The Global Cauchy Problem (n = 1)
2.8.1 The homogeneous case
2.8.2 Existence of a solution
2.8.3 The nonhomogeneous case. Duhamel’s method
2.8.4 Global maximum principle and uniqueness
2.8.5 Nonnegative solutions
2.8.6 The proof of the existence Theorem 2.12
2.9 An Application to Finance
2.9.1 European options
2.9.2 An evolution model for the price S
2.9.3 The Black-Scholes equation
2.9.4 The solutions
2.9.5 Hedging and self-financing strategy
2.10 Some Nonlinear Aspects
2.10.1 Nonlinear diffusion. The porous medium equation
2.10.2 Nonlinear reaction. Fisher’s equation
Problems
Chapter 3 The Laplace Equation
3.1 Introduction
3.2 Boundary value Problems. Uniqueness
3.3 Harmonic Functions
3.3.1 Discrete harmonic functions
3.3.2 Mean value properties
3.3.3 Maximum principles
3.3.4 The Hopf principle
3.3.5 The Dirichlet problem in a disc. Poisson’s formula
3.3.6 Harnack’s inequality and Liouville’s theorem
3.3.7 Analyticity of harmonic functions
3.4 Sub/Superharmonic Functions. The Perron Method
3.4.1 Sub/superharmonic functions
3.4.2 The method
3.4.3 Boundary behavior
3.5 A probabilistic solution of the Dirichlet problem
Recurrence and Brownian motion
3.6 Fundamental Solution and Newtonian Potential
3.6.1 The fundamental solution
3.6.2 The Newtonian potential
3.6.3 A divergence-curl system. Helmholtz decomposition formula
3.7 The Green Function
3.7.1 An integral identity
3.7.2 Green’s function
3.7.3 Green’s representation formula
3.7.4 The Neumann function
3.8 Uniqueness in Unbounded Domains
3.8.1 Exterior problems
3.9 Surface Potentials
3.9.1 The double and single layer potentials
3.9.2 The integral equations of potential theory
3.9.3 Proof of Theorem 3.47
Problems
Chapter 4 Scalar Conservation Laws and First Order Equations
4.1 Introduction
4.2 Linear Transport Equation
4.2.1 Pollution in a channel
4.2.2 Distributed source
4.2.3 Extinction and localized source
4.2.4 Inflow and outflow characteristics. A stability estimate
4.3 Traffic Dynamics
4.3.1 A macroscopic model
4.3.2 The method of characteristics
4.3.3 The green light problem
4.3.4 Traffic jam ahead
4.4 Weak (or Integral) Solutions
4.4.1 The method of characteristics revisited
4.4.2 Weak solutions
4.4.3 Piecewise smooth functions and the Rankine-Hugoniot condition
4.5 An Entropy Condition
4.6 The Riemann problem
4.6.1 Convex/concave flux function
4.6.2 Vanishing viscosity method. Travelling waves
4.6.3 Flux function with inflection points
4.7 The Method of Characteristics for Quasilinear Equations
4.7.1 Characteristics
4.7.2 The Cauchy problem
4.7.3 General solutions and Lagrange method of first integrals
4.7.4 Underground flow
4.8 Fully Nonlinear Equations
4.8.1 Characteristic strips
4.8.2 The Cauchy Problem
4.8.3 Complete integrals, general and singular solutions
Problems
Chapter 5 Waves and Vibration
5.1 General Concepts
5.1.1 Types of waves
5.1.2 Group velocity and dispersion relation
5.2 Transverse Waves in a String
5.2.1 The model
5.2.2 Energy
5.3 The One-dimensionalWave Equation
5.3.1 Initial and boundary conditions
5.3.2 Separation of variables
5.4 The d’Alembert Formula
5.4.1 The homogeneous equation
5.4.2 Generalized solutions and propagation of singularities
5.4.3 The fundamental solution
5.4.4 Nonhomogeneous equation. Duhamel’s method
5.4.5 Dissipation and dispersion
5.5 Second Order Linear Equations
5.5.1 Classification
5.5.2 Characteristics and canonical form
5.6 The Multi-dimensionalWave Equation (n > 1)
5.6.1 Special solutions
5.6.2 Initial boundary value problems. Uniqueness
5.7 Two Classical Models
5.7.1 Small vibrations of an elastic membrane
5.7.2 Small amplitude sound waves
5.8 The Global Cauchy Problem
5.8.1 Fundamental solution (n = 3) and strong Huygens’ principle
5.8.2 The Kirchhoff formula
5.8.3 The Cauchy problem in dimension 2
5.9 An Inverse Problem in Thermoacoustic Tomography
The mathematical model for the pressure
The inverse problem. The time reversal method
5.10 The Cauchy Problem with Distributed Sources
5.10.1 Retarded potentials (n = 3)
5.10.2 Radiation from a moving point source
5.11 Linear Water Waves
5.11.1 A model for surface waves
5.11.2 Dimensionless formulation and linearization
5.11.3 Deep water waves
5.11.4 Interpretation of the solution
5.11.5 Asymptotic behavior
5.11.6 The method of stationary phase
Problems
Chapter 6 Elements of Functional Analysis
6.1 Motivations
6.2 Norms and Banach Spaces
6.3 Hilbert Spaces
6.4 Projections and Bases
6.4.1 Projections
6.4.2 Bases
6.5 Linear Operators and Duality
6.5.1 Linear operators
6.5.2 Functionals and dual space
6.5.3 The (Hilbert) adjoint of a bounded operator
6.5.4 Ranges and Kernels. The inf-sup condition
6.6 Abstract Variational Problems
6.6.1 Bilinear forms. The Lax-Milgram Theorem
6.6.2 Approximation and Galerkin’s method
6.6.3 Minimization of quadratic functionals
6.6.4 The Ne.cas Theorem
6.7 Compactness and Weak Convergence
6.7.1 Compactness
6.7.2 Compactness in C(Ω) and in Lp (Ω)
6.7.3 Weak convergence and compactness
6.7.4 Compact operators
6.8 The Fredholm Alternative
6.8.1 Hilbert triplets
6.8.2 Solvability for abstract variational problems
6.8.3 The Fredholm alternative
6.9 Spectral Theory for Symmetric Bilinear Forms
6.9.1 Spectrum of a matrix
6.9.2 Separation of variables revisited
6.9.3 Spectrum of a compact self-adjoint operator
6.9.4 Application to abstract variational problems
6.10 Fixed Points Theorems
6.10.1 The Contraction Mapping Theorem
6.10.2 The Schauder Theorem
6.10.3 The Leray-Schauder Theorem
Problems
Chapter 7 Distributions and Sobolev Spaces
7.1 Preliminary Ideas
7.2 Test Functions and Mollifiers
7.3 Distributions
7.4 Calculus
7.4.1 The derivative in the sense of distributions
7.4.2 Gradient, divergence, Laplacian
7.5 Operations with Distributions
7.5.1 Multiplication. Leibniz rule
7.5.2 Composition
7.5.3 Division
7.5.4 Tensor or direct product
7.5.5 Convolution
7.6 Tempered Distributions and Fourier Transform
7.6.1 Tempered distributions
7.6.2 Fourier transform in
7.6.3 Fourier transform in
7.7 Sobolev Spaces
7.7.1 An abstract construction
7.7.2 The space H1(Ω)
7.7.3 The space H01(Ω)
7.7.4 The dual of H01(Ω)
7.7.5 The spaces Hm(Ω), m > 1
7.7.6 Calculus rules
7.7.7 Fourier transform and Sobolev spaces
7.8 Approximations by Smooth Functions and Extensions
7.8.1 Local approximations
7.8.2 Extensions and global approximations
7.9 Traces
7.9.1 Traces of functions in H1(Ω)
7.9.2 Traces of functions in Hm(Ω)
7.9.3 Trace spaces
7.10 Compactness and Embeddings
7.10.1 The Rellich Theorem
7.10.2 Poincar´e’s inequalities
7.10.3 Sobolev inequality in Rn
7.10.4 Bounded domains
7.11 Spaces Involving Time
7.11.1 Functions with values into Hilbert spaces
7.12 Sobolev spaces involving time
Problems
Chapter 8 Variational Formulation of Elliptic Problems
8.1 Linear Elliptic Equations
8.2 Notions of Solution
8.3 Problems for the Poisson Equation
8.3.1 Dirichlet problem
8.3.2 Neumann, Robin and mixed problems
8.3.3 Eigenvalues and eigenfunctions of the Laplace operator
8.3.4 An asymptotic stability result
8.4 General Equations in Divergence Form
8.4.1 Basic assumptions
8.4.2 Dirichlet problem
8.4.3 Neumann problem
8.4.4 Robin and mixed problems
8.5 Equilibrium of a Plate
8.6 Weak Maximum Principles
8.7 Regularity
Interior regularity
Global regularity
Mixed problems
Regularity in Lipschitz domains
Problems
Chapter 9 Weak Formulation of Evolution Problems
9.1 Parabolic Equations
9.2 The Cauchy-Dirichlet Problem for the Heat Equation
9.3 Abstract Parabolic Problems
9.3.1 Formulation
9.3.2 Energy estimates. Uniqueness and stability
9.3.3 The Faedo-Galerkin approximations
9.3.4 Existence
9.4 Parabolic PDEs
9.4.1 Problems for the heat equation
9.4.2 Regularity
9.4.3 General Equations
9.5 Weak Maximum Principles
9.6 The Wave Equation
9.6.1 Hyperbolic Equations
9.6.2 The Cauchy-Dirichlet problem
9.6.3 The method of Faedo-Galerkin
9.6.4 Solution of the approximate problem
9.6.5 Energy estimates
9.6.6 Existence, uniqueness and stability
Problems
Chapter 10 More Advanced Topics
10.1 The Linear Elastostatic System
10.2 The Stokes System
10.3 The Stationary Navier Stokes Equations
10.3.1 Weak formulation
10.3.2 Existence, and uniqueness for small data
10.4 A Control Problem
10.4.1 Structure of the problem
10.4.2 Existence and uniqueness of an optimal pair
10.4.3 Lagrange multipliers and optimality conditions
10.4.4 An iterative algorithm
10.5 Reaction-Diffusion Models
10.5.1 Dynamics of a population in a heterogeneous environment
10.5.2 A monotone iteration scheme
10.5.3 Long time behavior and steady states
10.5.4 Regularity
10.5.5 Persistence versus extinction
Problems
Chapter 11 Systems of Conservation Laws
11.1 Introduction
11.2 Linear Hyperbolic Systems
11.2.1 Characteristics
11.2.2 Classical solutions of the Cauchy problem
11.2.3 Homogeneous systems with constant coefficients. The Riemann problem
11.3 Quasilinear Conservation Laws
11.3.1 Characteristics and Riemann invariants
11.3.2 Weak (or integral) solutions and the Rankine-Hugoniot condition
11.4 The Riemann Problem
11.4.1 Rarefaction waves. Genuinely nonlinear systems
11.4.2 Solution of the Riemann problem by a single rarefaction wave
11.4.3 Lax entropy condition. Shock waves and contact discontinuities
11.4.4 Solution of the Riemann problem by a single k-shock
11.4.5 The linearly degenerate case
11.4.6 Local solution of the Riemann problem
11.5 The Riemann Problem for the p-system
11.5.1 Shock waves
11.5.2 Rarefaction waves
11.5.3 The solution in the general case
Problems
Appendix A Measures and Integrals
A.1 A Counting Problem
A.2 Measures and Measurable Functions
A.3 The Lebesgue Integral
A.4 Some Fundamental Theorems
A.5 Probability Spaces, Random Variables and Their Integrals
Appendix B Identities and Formulas
B.1 Gradient, Divergence, Curl, Laplacian
Orthogonal cartesian coordinates
Cylindrical coordinates
Spherical coordinates
B.2 Formulas
Gauss formulas
Identities
References
Partial Differential Equations
Mathematical Modelling
ODEs, Analysis and Functional Analysis
Numerical Analysis
Stochastic Processes and Finance
Index