Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

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ABSTRACT Smith 9622875 Smith will continue his research into the propagation of waves in nonhomogeneous mediums with ``rough'' sound speeds. The aim is to establish minimal differentiability assumptions on the metric determining the sound speed under which certain bounds on the solutions, known as the Strichartz estimates, hold. Under previous NSF funding, the investigator has shown that these bounds hold if the second order derivatives of the metric coefficients are bounded, and that this is the best possible condition of its sort that insures the validity of the Strichartz estimates. The proposed research includes replacing this condition by sharper, geometrically intrinsic ones. It is anticipated that the optimal condition is that the metric posess one bounded derivative, and in addition that the coefficients of the curvature tensor (certain combinations of second order derivatives) be bounded. The investigator is also continuing research into the development of function spaces adapted to oscillatory integral and hyperbolic problems. Previous NSF funding supported the development of a family of Hardy spaces adapted to fixed-time estimates for solutions of the wave equation. The proposed research includes developing similar spaces adapted to controlling time-averaged norms of solutions. The study of waves travelling in media with rough sound speeds (that is, where the speed may vary sharply from one point to another) is of both theoretical and practical significance. The theoretical interest comes from nonlinear equations, such as the Einstein equations for gravitation, in which the wave speed depends on the solution under consideration. Since rough solutions arise naturally, one needs to understand how waves travel in rough media to show that solutions exist, and to understand their properties. Physics predicts that energy should travel along geodesics, curves that represent the shortest path between points. Our research is sh owing that this is true even for rough sound speeds, under essentially the weakest assumptions that assure that geodesics exist and are well defined. The practical significance is that our research yields a mathematical construction for solving the wave equation, which is valid for rough sound speeds and stable under perturbations of the medium. It is clear that a computational algorithm based on such a construction will be inherently more stable than one requiring more restrictive conditions on the media. Our research would help establish that such algorithms are both stable and convergent.