This book addresses the issue of uniqueness of a solution to a problem – a very important topic in science and technology, particularly in the field of partial differential equations, where uniqueness guarantees that certain partial differential equations are sufficient to model a given phenomenon.
This book is intended to be a short introduction to uniqueness questions for initial value problems. One often weakens the notion of a solution to include non-differentiable solutions. Such a solution is called a weak solution. It is easier to find a weak solution, but it is more difficult to establish its uniqueness. This book examines three very fundamental equations: ordinary differential equations, scalar conservation laws, and Hamilton-Jacobi equations. Starting from the standard Gronwall inequality, this book discusses less regular ordinary differential equations. It includes an introduction of advanced topics like the theory of maximal monotone operators as well as what is called DiPerna-Lions theory, which is still an active research area. For conservation laws, the uniqueness of entropy solution, a special (discontinuous) weak solution is explained. For Hamilton-Jacobi equations, several uniqueness results are established for a viscosity solution, a kind of a non-differentiable weak solution. The uniqueness of discontinuous viscosity solution is also discussed. A detailed proof is given for each uniqueness statement.
The reader is expected to learn various fundamental ideas and techniques in mathematical analysis for partial differential equations by establishing uniqueness. No prerequisite other than simple calculus and linear algebra is necessary. For the reader’s convenience, a list of basic terminology is given at the end of this book.
Author(s): Mi-Ho Giga , Yoshikazu Giga
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 155
Tags: Monotone Operator, DiPerna-Lions Theory, Conservation Laws, Weak Solution, Hamilton-Jacobi Equations, Entropy Solution, Viscosity Solution
Preface
Contents
1 Uniqueness of Solutions to Initial Value Problems for Ordinary Differential Equations
1.1 Gronwall-Type Inequalities and Uniqueness of Solutions
1.1.1 Lipschitz Condition
1.1.2 Gronwall Inequality
1.1.3 Osgood Condition
1.1.4 A Solenoidal Vector Field Having Bounded Vorticity
1.1.5 Equation with Fractional Time Derivative
1.2 Gradient Flow of a Convex Function
1.2.1 Maximal Monotone Operator and Unique Existence of Solutions
1.2.2 Canonical Restriction
1.2.3 Subdifferentials of Convex Functions
1.2.4 Gradient Flow of a Convex Function-Energetic Variational Inequality
1.2.5 Simple Examples
1.3 Notes and Comments
1.3.1 More on Uniqueness on Continuous Vector Field b
1.3.2 Forward Uniqueness on Discontinuous Vector Field b
1.3.3 Estimates of Lipschitz Constant
1.3.4 A Few Directions for Applications
1.4 Exercises
2 Ordinary Differential Equations and Transport Equations
2.1 Uniqueness of Flow Map
2.2 Transport Equations
2.3 Duality Argument
2.4 Flow Map and Transport Equation
2.5 Notes and Comments
2.6 Exercises
3 Uniqueness of Solutions to Initial Value Problems for a Scalar Conversation Law
3.1 Entropy Condition
3.1.1 Examples
3.1.2 Formation of Singularities and a Weak Solution
3.1.3 Riemann Problem
3.1.4 Entropy Condition on Shocks
3.2 Uniqueness of Entropy Solutions
3.2.1 Vanishing Viscosity Approximations and Entropy Pairs
3.2.2 Equivalent Definition of Entropy Solution
3.2.3 Uniqueness
3.3 Notes and Comments
3.4 Exercises
4 Hamilton–Jacobi Equations
4.1 Hamilton–Jacobi Equations from Conservation Laws
4.1.1 Interpretation of Entropy Solutions
4.1.2 A Stationary Problem
4.2 Eikonal Equation
4.2.1 Nonuniqueness of Solutions
4.2.2 Viscosity Solution
4.2.3 Uniqueness
4.3 Viscosity Solutions of Evolutionary Hamilton–Jacobi Equations
4.3.1 Definition of Viscosity Solutions
4.3.2 Uniqueness
4.4 Viscosity Solutions with Shock
4.4.1 Definition of Semicontinuous Functions
4.4.2 Example for Nonuniqueness
4.4.3 Test Surfaces for Shocks
4.4.4 Convexification
4.4.5 Proper Solutions
4.4.6 Examples of Viscosity Solutions with Shocks
4.4.7 Properties of Graphs
4.4.8 Weak Comparison Principle
4.4.9 Comparison Principle and Uniqueness
4.5 Notes and Comments
4.5.1 A Few References on Viscosity Solutions
4.5.2 Discontinuous Viscosity Solutions
4.6 Exercises
5 Appendix: Basic Terminology
5.1 Convergence
5.2 Measures and Integrals
Bibliography
Index
