An analytical and numerical study of steady patches in the disc

TL;DR

This paper proves the existence of rotating vortex patches in a disc for Euler equations, exploring boundary effects and dynamics for different symmetries through analytical and numerical methods.

Contribution

It establishes the existence of m-fold rotating patches in a disc, including simply- and doubly-connected cases, with new insights into boundary interactions and symmetry effects.

Findings

Existence of m-fold rotating patches proven analytically.

Numerical experiments show boundary-patch interactions.

Rich dynamics for symmetries m=1 and m=2 due to boundary effects.

Abstract

In this paper, we prove the existence of $m$-fold rotating patches for the Euler equations in the disc, for both simply-connected and doubly-connected cases. Compared to the planar case, the rigid boundary introduces rich dynamics for the lowest symmetries $m=1$ and $m=2$. We also discuss some numerical experiments highlighting the interaction between the boundary of the patch and the rigid one.