The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations
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Summary
Despite decades of research and progress in the theory of generalized solutions to first-order nonlinear partial differential equations, a gap between the local and the global theories remains: The Cauchy characteristic method yields the local theory of classical solutions. Historically, the global theory has principally depended on the vanishing viscosity method.
The authors of this volume help bridge the gap between the local and global theories by using the characteristic method as a basis for setting a theoretical framework for the study of global generalized solutions. That is, they extend the smooth solutions obtained by the characteristic method.
The authors offer material previously unpublished in book form, including treatments of the life span of classical solutions, the construction of singularities of generalized solutions, new existence and uniqueness theorems on minimax solutions, differential inequalities of Haar type and their application to the uniqueness of global, semi-classical solutions, and Hopf-type explicit formulas for global solutions. These subjects yield interesting relations between purely mathematical theory and the applications of first-order nonlinear PDEs.
The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations represents a comprehensive exposition of the authors' works over the last decade. The book is self-contained and assumes only basic measure theory, topology, and ordinary differential equations as prerequisites. With its innovative approach, new results, and many applications, it will prove valuable to mathematicians, physicists, and engineers and especially interesting to researchers in nonlinear PDEs, differential inequalities, multivalued analysis, differential games, and related topics in applied analysis.
Table of Contents
Preface
Local Theory on Partial Differential Equations of First Order
Characteristic Method and Existence of Solution
A Theorem of A. Haar
A Theorem of T. Wazewski
Life Spans of Classical Solutions of Partial Differential Equations of First Order
Introduction
Life Spans of Classical Solutions
Global Existence of Classical Solutions
Behavior of Characteristic Curves and Prolongation of Classical Solutions
Introduction
Examples
Prolongation of Classical Solutions
Sufficient Conditions for Collision of Characteristic Curves I
Sufficient Conditions for Collision of Characteristic Curves II
Equations of Hamilton-Jacobi Type in One Space Dimension
Non-Existence of Classical solutions and Historical Remarks
Construction of Generalized Solutions
Semi-Concavity of Generalized Solutions
Collision of Singularities
Quasi-Linear Partial Differential Equations of First Order
Introduction and Problems
Difference Between Equations of Conservation Law and Equations of Hamilton-Jacobi Type
Construction of Singularities of Weak Solution
Entropy Condition
Construction of Singularities for Hamilton-Jacobi Equations in Two Space Dimensions
Introduction
Construction of Solution
Semi-Concavity of the Solution u = u(t,x)
Collision of Singularities
Equations of Conservation Law without Convexity Condition in One Space Dimension
Introduction
Rarefaction Waves and Contact Discontinuity
An Example of an Equation of Conservation Law
Behavior of the Shock S1
Behavior of the Shock S2
Differential Inequalities of Haar Type
Introduction
A Differential Inequality of Haar Type
Uniqueness of Global Classical Solutions to the Cauchy Problem
Generalizations to the Case of Weakly-Coupled Systems
Hopf's Formulas for Global Solutions of Hamilton-Jacobi Equations
Introduction
The Cauchy Problem with Convex Initial Data
The Case of Nonconvex Initial Data
Equations with Convex Hamiltonians f = f(p)
Hopf-Type Formulas for Global Solutions in the Case of Concave-Convex Hamiltonians
Introduction
Conjugate Concave-Convex Functions
Hopf-Type Formulas
Global Semiclassical Solutions of First-Order Partial Differential Equations
Introduction
Uniqueness of Global Semiclassical Solutions to the Cauchy Problem
Existence Theorems
Minimax Solutions of Partial Differential Equations with Time-Measurable Hamiltonians
Introduction
Definition of Minimax Solutions
Relations with Semiclassical Solutions
Invariance of Definitions
Uniqueness and Existence of Minimax Solutions
The Case of Monotone Systems
Mishmash
Hopf's Formulas and Construction of Global Solutions via Characteristics
Smoothness of Global Solutions
Relationship Between Minimax and Viscosity Solutions
Appendix I: Global Existence of Characteristic Curves
Appendix II: Convex Functions, Multifunction, and Differential Inclusions
References
Index
