The Characteristic Method and its Generalizations for First-order Nonlinear Partial Differential Equations

The Characteristic Method and its Generalizations for First-order Nonlinear Partial Differential Equations

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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 The authors offer material previously unpublished in book form, including treatments of the life span of classical solutions, the construction of singularities of generalized solutions, new existence and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 more

Product details

  • Hardback | 256 pages
  • 157.5 x 236.2 x 20.3mm | 557.93g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 1584880163
  • 9781584880165

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 Indexshow more