Mathematical Sciences: Size and Regularity Estimates for Solutions to Hyperbolic Equations

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9401819 Beals The work supported by this award involves the study of several questions on properties of solutions to strictly hyperbolic partial differential equations. The equations are an outgrowth of the classical wave equation which lies at the heart of the theory of hyperbolic equations. Refined estimates in the linear case will be derived, to analyze the properties of solutions to the physically more realistic nonlinear problems. Among the questions to be considered are: estimates of time decay for appropriate measures of the size of a solution and analysis of nonconstant coefficient problems corresponding to nonuniform media in which wave propagation takes place. Such estimates will also be considered for solutions to wave equations on domains with boundary, such as in the exterior of an obstacle, to find the best possible time decay subject to the weakest restrictions on the data. Finally, smoothness properties of solutions to nonlinear wave equations as determined by the corresponding properties of the data will be treated. The general propagation of smoothness in a wave satisfying a nonlinear equation will be reconsidered in a setting of less initial smoothness. Examples will be considered where the formation of nonlinear singularities develop in cases in which nonsmooth surfaces appear, both for interior domains and in the presence of a boundary. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***