Harmonic Analysis and Hyperbolic Partial Differential Equations

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DMS-9970407AbstractThe investigator is continuing his research into the behavior ofwave propagation in nonhomogeneous media with "rough" wave speeds.The goal is to obtain estimates of Strichartz and nullform typeon solutions to variable coefficient wave equations underminimal differentiability assumptions on the coefficients.Under current NSF funding, the investigator has shown thatthe Strichartz estimates hold if the curvature tensor of the wavespeed metric is bounded and measurable. The proposed researchincludes extending these results to obtain estimates for bilinearnull-forms of solutions, as well as estimates for eigenfunctionsof the Laplacian on spaces with bounded curvature. The investigatoris also involved in joint work studying the Strichartz and bilinearnullform estimates for waves which reflect from convex obstacles.Previous joint research has obtained such estimates in bounded regions of space-time. The proposed research involves combining these local methods with known energy decay estimates to obtain suchestimates valid globally in space and time. These results are thenbeing applied to establish long time existence for certain nonlinearwave equations on the region exterior to a convex obstacle.The development of techniques to handle partial differential equationswith low regularity coefficients is of both theoretical and practicalinterest. The theoretical interest comes from the study ofnonlinear equations in math and physics, such as Einstein's equationsfor the gravitational field, or the equations of fluid dynamics. A central question for such equations is the "long time" existenceof solutions; that is, showing that solutions do not blow upat some future time. The main technique for doing this is to showthat one can control the size of the nonlinear part of the equationin terms of quantities that are known to remain controlled, suchas the energy of the solution. For Einstein's equations, thisrequires understanding the wave equation with rough wave speeds.The practical importance of such studies is that a theoreticalunderstanding of the propagation of waves in rough media shouldlead to more efficient algorithms for the numerical computationof solutions. Our work involves splitting the solution intoelementary pieces, each of which behaves in a particularly simplemanner. This is analogous to the decomposition of functions intowavelets, much of the modern theory of which was developed tounderstood nonlinear equations of stable-state phenomena. Waveletshave in turn led to simple and stable algorithms for solving such equations, as well as to powerful algorithms in the fields of imageand signal processing.