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Home » Model problems in fluid dynamics: Instability via degenerate dispersion for generalized surface quasi-geostrophic models with singular velocities

Seminar

Model problems in fluid dynamics: Instability via degenerate dispersion for generalized surface quasi-geostrophic models with singular velocities March 08, 2021 (08:30 AM PST - 09:30 AM PST)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Sung-Jin Oh (University of California, Berkeley)
Description

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

Video

Instability Via Degenerate Dispersion for Generalized Surface Quasi-Geostrophic Models with Singular Velocities

Abstract/Media

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

Abstract: The primary purpose of this talk is to elucidate an instability mechanism, which will be referred to as degenerate dispersion, that leads to illposedness of the Cauchy problem in arbitrarily high-regularity Sobolev spaces for a number of nonlinear PDEs of hydrodynamics and magnetohydrodynamics (MHD) that respect conservation of energy. Due to the conservation structure, the instability mechanism is necessarily different from that of, say, the reverse heat equation; rather, it is a mechanism by which energy gets concentrated into small scales at an arbitrarily fast rate due to the degeneration of the dispersion relation. 

In this talk, I will focus on generalized surface quasi-geostrophic (gSQG) models with singular velocities. I will give a heuristic description of the phenomenon via geometric-optical ideas (or classical-quantum correspondence), and then discuss the mathematical tools recently developed to capture this phenomenon in the nonlinear setting. This talk is based on joint works with Dongho Chae and In-Jee Jeong.

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Instability Via Degenerate Dispersion for Generalized Surface Quasi-Geostrophic Models with Singular Velocities