Mathematical Sciences: Global Properties of Nonlinear Hyperbolic Equations Arising in Mathematical Physics

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This project continues work of the principal investigator on problems related to nonlinear partial differential equations arising in mathematical physics. The equations are typically of hyperbolic type. Objectives of this work include the development of methods for proving global existence, stability and asymptotic behavior results for classical field equations. Among the problems to be considered are the global nonlinear stability of Minkowski, Schwartzchild and Kerr solutions of the Einstein vacuum equations. Also the asymptotic behavior of the Yang-Mills equations for general initial conditions will be studied. Work will also be done on establishing global existence and asymptotic results for the Maxwell-Dirac equations subject to small initial conditions. A second line investigation involves analysis of the formation of singularities for quasilinear wave equations in three space dimensions. It has been known for some time that solutions to certain classes of equations have developed singularities in finite time . Very little is known about the nature of these singularities near the onset of blow up and the three dimensional case is the least understood of all. One of the difficulties to be overcome is conceiving a means for visualizing solutions whose singularities only occur after exponentially long periods of time have elapsed. Work will continue toward developing new methods for the study of global smooth solutions for semilinear wave equations. In wave equations where the nonlinear term is a power of the unknown function, it is known that solutions remain smooth indefinitely if the power is less than five. For higher powers almost nothing is known, although it is believed that some version of singular local energy estimates will provide the information necessary for the resolution of this problem. In addition to natural applications to mathematical physics, this work will provide methods which will be applicable to the analysis of large classes of differential equations.