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Home » Water waves and other interface problems (Part 2): Fronts Solutions of the Surface Quasi-Geostrophic Equation

Seminar

Water waves and other interface problems (Part 2): Fronts Solutions of the Surface Quasi-Geostrophic Equation February 23, 2021 (09:30 AM PST - 10:30 AM PST)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Jingyang Shu (Temple University)
Description

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Video

Fronts Solutions of the Surface Quasi-Geostrophic Equation

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Abstract: 

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. It is shown that in some cases, solutions of the two-front equations are not linearly stable. Numerical simulations for front solutions suggest the formation of finite-time singularity. This is joint work with John K. Hunter and Qingtian Zhang. 

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Fronts Solutions of the Surface Quasi-Geostrophic Equation