The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations (Chapman and Hall /Crc Monographs and Surveys in Pure and Applied Mathematics)

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 The authors offer material previously unpublished in book form, including treatments of the life span of classical solutions, the construction of singularities of generalized solutions, new existence and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. The authors present a comprehensive exposition -- and updated revisions of their work over the last decade -- of developments in the characteristic method for the theory of generalized solutions to the Cauchy problem first-order nonlinear partial differential equations. This book fills a gap between the local theory obtained by the characteristic method and the global theory that principally depends on the vanishing viscosity method. Partial differential equations of first-order have been studied from various points of view: for example, classical mechanics, variational method, geometrical optics, etc.