Mathematical Sciences: Existence and Blow-up of Solutions of Nonlinear Wave Equations

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Work to be done on this project concerns the time evolution of solutions to partial differential equations, especially nonlinear wave equations. The basic question which arises in these studies is the possibility of breakdown or blow-up of solutions. For some equations one has global existence of solutions whereas for others the solution will fail to exist after some finite time. One focus of the work is in the study of low regularity solutions, that is, solutions with low regularity data. The motivation is from equations in physics in which energy is conserved. If local existence can be established, the conserved energy will guarantee existence in the energy norm indefinitely. Present results require that a large number of derivatives be small, leaving no sense of what can go wrong. In addition to studying the period of existence of solutions, work will also be done in trying to describe the behavior of solutions at the onset of blow-up, perhaps one of the most challenging problems facing researchers in this area today. Many important equations in physics can be written as systems of nonlinear wave equations. These include classical field theory in a fixed gauge, continuum mechanics and the classical equations of Einstein, Yang Mills and equations of nonlinear elasticity. The mathematical study of how the solutions to these systems progress with time centers on some of the most important questions under investigation in mathematical analysis.