Bifurcation for hollow vortex desingularization

Series
PDE Seminar
Time
Tuesday, October 1, 2024 - 3:30pm for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ming Chen – University of Pittsburgh – mingchen@pitt.eduhttps://sites.pitt.edu/~mingchen/
Organizer
Gong Chen

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; we can think of it as a spinning bubble of air in water. In this talk, we present a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. The machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we obtain the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve. This is a joint work with Samuel Walsh (Missouri) and Miles Wheeler (Bath).