Regularity Properties of Nonlinear Evolution Equations

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ABSTRACTThe main goal of the proposal is to develop new analytic methods to deal with the fundamental issue of global regularity for Nonlinear Wave Equations such as Wave Maps, Yang-Mills and the Einstein Vacuum Equations. We propose two specific problems to investigate. The first concerns the critical global well-posedness for Wave Maps and the second is to establish local well posedness in $H^2$ for the Einstein Vacuum equations.Nonlinear Wave equations are at the heart of some of our basic physical theory such as General Relativity, Electrodynamics, Elasticity etc. Despite a lot of progress made throughout last century our knowledge of nonlinear waves remains rudimentary. The proposal outlines some directions in which we expect to make considerable progress. The main one concerns the Einstein field equations. It is well known that these equations can develop black holes and singularities. This fact makes it imperative to develop a theory for rough solutions for these equations. The Wave Maps equations can be viewed as a simplified model problem for the Einstein equations. We hope that a better understanding of rough solutions of these equations will help us make progress in connection to the former equations also.