Michael Beals

Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems - Progress in Mathematics Michael Beals 1989 edition

Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems presents the first expository overview of the use of microlocal analysis and commutator techniques in the study of solutions to nonlinear wave equations. In a clear, comprehensive manner the author synthesizes a variety of research results from the last ten years, providing detailed proofs and descriptions of the central ideas behind the arguments and includes a number of figures to illustrate the theory's geometric aspects. The linear theory of the microlocal propagation of singularities and regularity is briefly recalled to motivate the nonlinear study. Weak singularities which propagate along null bicharacteristics are analyzed, and examples which exhibit singularities not present in the linear case are constructed. A selection of recent results on problems on the interior of a domain and on domains with boundary is presented. The book will be an invaluable reference for active researchers in linear microlocal analysis, hyperbolic equations, pseudo-differential operators, and nonlinear wave equations. It should also make excellent supplemental reading for a second year graduate course in partial differential equations.