Proposal: DMS-9734866 Principal Investigator: Christopher D. Sogge Abstract: Sogge will work on various problems in Fourier analysis and nonlinear wave equations. He would like to concentrate mainly on two types of problems. First, he is interested in proving local and global existence theorems for nonlinear wave equations in various settings, including wave equations outside of convex obstacles. For this, the main analysis involves proving a suitable "Strichartz estimate." Such estimates in the situation where there is a boundary are much more delicate than in the case without boundary; however, some preliminary results have already been obtained in joint work with Hart Smith. Sogge is also very interested in studying maximal averages over curves and especially interested in trying to establish such estimates in the curved space setting. Sogge has been intrigued for some time by certain questions in nonlinear partial differential equations and Fourier analysis. Such issues arise naturally in many contexts. For instance, the equations of general relativity are just nonlinear wave equations. One of the main things Sogge would like to do is to show that such equations have solutions in the region outside of an obstacle. This might be thought of as a model for equations describing black holes, where the obstacle plays the role of the black hole. Sogge is also interested in basic problems in Fourier analysis. Since its introduction in the pioneering work of Fourier himself on problems of heat conduction, Fourier analysis has been the main tool to study differential equations, which constitute the formal language of physics.
