Steady vortex patches on flat torus with a constant background vorticity

TL;DR

This paper constructs steady vortex patch solutions on a flat torus with constant background vorticity, addressing unique challenges posed by the domain's topology and invariance, and proving smoothness and convexity of patch boundaries.

Contribution

It introduces a novel method for constructing vortex patches on a flat torus with background vorticity, overcoming domain and invariance difficulties, and establishes boundary regularity and convexity.

Findings

Constructed vortex patch solutions on flat torus with background vorticity.

Proved $C^ abla$ regularity and convexity of patch boundaries.

Addressed challenges due to translational invariance and domain topology.

Abstract

We construct a series of vortex patch solutions in a doubly-periodic rectangular domain (flat torus), which is accomplished by studying the contour dynamic equation for patch boundaries. We will illustrate our key idea by discussing the single-layered patches as the most fundamental configuration, and then investigate the general construction for $N$ patches near a point vortex equilibrium. Different with the case of bounded domains in $\mathbb R^2$, a constant background vorticity will arise from the compact nature of flat torus, and the $2$-dimensional translational invariance will bring troubles on determining patch locations. To overcome these two difficulties, we will add additional terms for background vorticity and introduce a centralized condition for location vector. By utilizing the regularity difference of terms in contour dynamic equations, we also obtain the $C^\infty$ regularity and convexity of boundary curves.