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Sharp Fronts for the SQG Equation

PDE and Applied Math Seminar

Speaker: Jingyang Shu, Temple University
Location: ZOOM
Start time: Fri, May 1 2020, 4:10PM

Piecewise-constant fronts of the surface quasi-geostrophic (SQG) equation support surface waves. For planar SQG fronts, the formal contour dynamics equation does not converge. We use a decomposition method to overcome this difficulty and obtain a well-formulated meaningful contour dynamics equation for fronts that are described as a graph. The resulting equation is a nonlocal quasi-linear equation with logarithmic dispersion. With smallness and smoothness assumptions on the initial data, the solutions exist and are global. For two SQG fronts, the contour dynamics equations form a system with more complicated dispersion relations as well as quadratic nonlinearities. Numerical simulations for front solutions suggest the formation of finite-time singularity. This is joint work with John K. Hunter and Qingtian Zhang.