Bilinear Estimates in Analysis and Partial Differential Equations

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This research project concerns bilinear Fourier analysis. Broadly speaking, Fourier analysis is a mathematical discipline for the study of signals, such as sound and images. The study of signals by way of Fourier analysis involves breaking them down into fundamental pieces that are less complex and, therefore, easier to examine. Information obtained from the individual pieces is then synthesized to obtain information about the original signal. Fourier analysis has had far-reaching applications in other areas of mathematics, physics, engineering, medicine, industry, and the applied sciences. This project will investigate central questions in the field of bilinear Fourier analysis, where a pair of signals are analyzed simultaneously. The outcomes are anticipated to have applications in the theory of partial differential equations, to topics as diverse as fluid dynamics, quantum mechanics, and optics. The project will also contribute to the integration of research and education at the graduate and undergraduate levels.The project aims to contribute to new developments in bilinear Fourier analysis through the investigation of a suite of interrelated questions motivated by applications to analysis and partial differential equations. The project will investigate several approaches, based on tools including representations of functions, Littlewood-Paley techniques, and symbolic calculus, to generate a host of new bilinear estimates and boundedness properties of bilinear pseudodifferential operators. The results are expected to apply to the pointwise multiplication properties of function spaces, local well-posedness results for the Euler equations and the ideal magnetohydrodynamic equations, and scattering properties of solutions of systems of partial differential equations associated to local and nonlocal operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.