Mathematical Sciences: Geometric Methods and Nonlinear Hyperbolic Equations

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New geometrical methods will be used in the analysis of two problems concerning nonlinear hyperbolic equations. One thrust will focus on the establishment of a new decay estimate for the linear nonhomogeneous Klein-Gordon equation in three space dimensions. Decay estimates arise when one studies linear problems associated with nonlinear hyperbolic equations. This approach has been suggested by successful applications by Klainerman and Hormander to quadratically nonlinear wave equations. A similar program is planned for the Klein-Gordon equation where uniform decay estimates will be sought. The solution is to be bounded by an appropriate norm of the forcing function - which is not assumed to have compact support. When this program is carried out, it should be possible to give a direct proof of the existence of global small solutions of the equation. A second line of investigation continues earlier work on obtaining lower bounds for the life-span of smooth three- dimensional compressible flows. These ideal, compressible flows are considered with small initial disturbances that are smooth and localized. Near the front of the ensuing disturbance a singularity or discontinuity is eventually detected. These are shocks in the acoustical waves and are predictable. The thrust of this effort is to determine whether or not singularities can form at an earlier time and, if so, to characterize their nature. The functional relationship between the solution and its initial configuration is expected to be an exponential involving the size of the initial support. This work will have natural application to studies of wave motion and fluid dynamics.