Abstract

Vortex dynamics is a model of interacting particles in a potential field with logarithmic singularities. It is not only an intrinsic theoretical extension in the field of classical fluid mechanics, but it would also be applicable to modern physics such as quantum mechanics and flows of superfluid films. We consider the equilibrium states of these vortex structures. The method of complex analysis plays a significant role in constructing vortex equilibria. After my stay on the occasion of the CAT program at Isaac Newton Institute in 2019, many international collaborations were initiated, and we have thus obtained more new results on vortex equilibria. In this talk, I will provide these recent results: First, we consider point vortex equilibria on the surface of a curved torus and a flat torus (i.e., a doubly periodic plane). They are embedded in the background smooth vorticity distributions such as a constant vorticity distribution and a Liouville-type vorticity distribution, where the vorticity is represented by an exponential function of the stream function. Second, we show the existence of a family of equilibrium states involving different members of straight vortex sheets rotating about a center of rotation and with endpoints at the vortices of a regular polygon.