Sharp Fronts for the SQG Equation

Sharp Fronts for the SQG Equation

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Jingyang Shu, University of California, Davis

Zoom link:

https://princeton.zoom.us/j/9148065146

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 front equation admits global solutions. For two SQG fronts, the contour dynamics equations form a system with more complicated dispersion relations as well as nonlinear interactions between the two fronts. Numerical simulations for front solutions suggest the formation of finite-time singularity. This is joint work with John K. Hunter and Qingtian Zhang.