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