Desingularization and global continuation for hollow vortices #

Miles Wheeler, Robin Ming Chen, Samuel Walsh

12:30 Monday in 2Q42.

Part of the Advances in water waves and free-surface flows session.

Abstract #

Hollow vortices are regions of constant pressure with finite circulation embedded into an otherwise irrotational flow. We show quite generally that configurations of hollow vortices which steadily translate or rotate – sometimes called ‘H-states’ – can be rigorously constructed by desingularizing analogous configurations of point vortices. While the resulting solutions are not given explicitly, we provide the leading-order terms in an asymptotic expansion. Using global bifurcation theory, we then go on to describe how these curves of solutions can be extended until a singularity is encountered. As applications, we give what appear to be the first rigorous existence results for rotating hollow vortex pairs and for stationary hollow vortex tripoles.