High frequency scattering with applications to fluids

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The proposed project is part of the primary supervisor's broader research program on spectral and scattering theory. This project focuses on the study of high frequency solutions to wave-type equations. The student's work will focus on the study of scattered waves in the presence of trapping. It has been known since the work of Burq that estimates for such waves must suffer losses relative to non-trapping situations. However, the precise energies at which these losses occur are not fully understood. This project will aim to improve our understanding of this phenomena. While undertaking the project, the student will be trained to use several important theories in mathematics. The student will acquire a strong foundation in mathematical scattering theory as well as in microlocal analysis and functional analysis. These tools have recently found applications in an enormous variety of problems including the study of dynamical systems, internal forced waves in fluids, dynamics of free boundary problems, and general relativity to name a few. The precise understanding of the linearised dynamics is of particular importance in the context of nonlinear PDE. In the course of the project, the student will be trained to look for and implement these tools, particularly in the context of mathematical fluid dynamics; an area of expertise of the secondary supervisor.Research Area: Mathematical analysis