Applied mathematics in vorticity and vortex dynamics

Abstract: Vorticity and vortex dynamics is an important subfield of fluid dynamics. It is an interdisciplinary science at the intersection of mathematics, physics and engineering. I will start this talk by providing a brief overview of the different problems of interest within vortex dynamics, with an emphasis on various mathematical approaches to these problems. I will then introduce a couple of problems that I have worked on, namely compressible vortex dynamics and exact solutions of the steady incompressible Euler equation, both in two-dimensions. Some recent results in these topics include a formula for the correction due to compressibility to the speed of the incompressible von Karman point vortex street, and an analysis of finite-area effects when the point vortices are replaced by hollow vortices which results in a free boundary problem. We have uncovered new exact solutions of the steady incompressible Euler equation, which we call "Liouville links and chains". These involve point vortices embedded in a background sea of smooth vorticity described by the Liouville PDE; examples include iterated solutions given in terms of the Adler--Moser polynomials arising in KdV theory. After briefly describing these solutions, I will end by discussing some current and future projects on related topics.