Nonlinear Wave Progatation

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proposal 9801558 NONLINEAR WAVE PROPAGATION PROJECT SUMMARY Markus Keel A difficulty common to many interesting nonlinear evolution equations is so-called L-p ill-posedness. That is, the corresponding linear time evolutions cannot be effectively compared at each fixed time to the initial data using a single L-p space (Lebesgue space with exponent p) when p is not equal to 2. (When p = 2, there is often a conservation of energy which provides just such an estimate.) This obstacle is present in equations ranging from classical field theory to the higher dimensional analogues of hyperbolic systems dealt with in 1 space dimension by Glimm's method. We propose to study two analytical methods developed to deal with L-p ill-posedness in the context of nonlinear wave equations and nonlinear Schrodinger equations. The first is space-time estimates for the corresponding linear problems. We propose to study in collaboration with T. Tao the endpoint Strichartz estimates for the Schrodinger and wave equation in space dimensions 2 and 3, respectively. This would be an extension of previous work by Tao and the author. We propose second to study generalized Sobolev estimates and the method of conformal compactification. We will be looking, in collaboration with C. Sogge and H. Smith, at the specific problem of global in time existence for a system of quasilinear wave equations with small initial data in the exterior of an obstacle. Finally, we would like to bring these techniques to the question of how the choice of gauge affects regularity of the solution in the context of the hyperbolic Yang-Mills equations. The physics behind many interesting systems currently being investigated (for example, fluid flow or the LIGO gravitational wave detectors) leads scientists to a number of very difficult questions in nonlinear mathematics. When we consider the nonlinear partial differential equations arising in fluid mechanics and general relativity, some natural questions are "how big is a solution at long times" or even "does a solution exist for all times". We have only very incomplete answers to these questions. The present proposal aims to further understand promising techniques in nonlinear hyperbolic pde which might eventually be brought to physical systems. For example, the question above regarding endpoint Strichartz estimates may shed light on the decay properties of waves. The obstacle problem mentioned above is a toy model for the stability of the Schwarzschild space in general relativity.