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Singular flows, zeroth order pseudodifferential operators and spectra

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The propagation of internal gravity waves in stratified media (such as those found in ocean basins and lakes) leads to the development of attractors. These structures accumulate much of the wave energy and can make the fluid flow highly singular. These questions have been the subject of fascinating recent analytical developments by de Verdiere & Saint-Raymond, and Zworski and co-workers, who examine a simplified model which retains many of the important features. These are related to a certain zeroth-order pseudodifferential operator.

In this talk, we first review the physical phenomenon, and the (highly simplified) model evolution problem. We next describe a high-accuracy computational method to solve the evolution problem, whose long-term behaviour is known to be non-square-integrable. Then, we use similar tools to discretize the corresponding eigenvalue problem. Since the eigenvalues are embedded in a continuous spectrum, their computation is based on viscous approximations. We also study the long-term evolution of the dynamics of the system. This is joint work with Javier Almonacid.

This talk is part of the Applied and Computational Analysis series.

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