Steady vortex patches on flat torus with a constant background vorticity

Takashi Sakajo; Changjun Zou

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arXiv · 2501.04271

Steady vortex patches on flat torus with a constant background vorticity

Takashi Sakajo·Changjun Zou

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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.

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Takashi Sakajo, Changjun Zou. 2025-01-08. Steady vortex patches on flat torus with a constant background vorticity. https://arxiv.org/abs/2501.04271

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