Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Steady vortex patches on flat torus with a constant background vorticity

Takashi Sakajo, Changjun Zou

TL;DR

The paper addresses steady vortex patches for the 2D Euler equation on a doubly periodic domain (flat torus) with a constant background vorticity. It develops a contour-dynamics framework, derives a precise Green-function expansion on $\mathbb{T}^2$, and constructs both single-layered and multi-patch configurations near point-vortex equilibria by introducing a centralized center condition and an adjustable background strength. Using a nonlinear implicit-function theorem together with a bootstrap argument, the authors prove the existence of $C^\infty$-regular, convex patch boundaries and extend the construction to general $N$-patch lattices. The work provides a rigorous route to patch lattices in a compact, translationally-invariant setting and enhances understanding of near-equilibrium vortex structures in 2D turbulence. Key techniques include contour dynamics, Green-function analysis on the torus, Kirchhoff-Routh functionals, and a careful linearization/regularity bootstrap strategy.

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.

Steady vortex patches on flat torus with a constant background vorticity

TL;DR

The paper addresses steady vortex patches for the 2D Euler equation on a doubly periodic domain (flat torus) with a constant background vorticity. It develops a contour-dynamics framework, derives a precise Green-function expansion on , and constructs both single-layered and multi-patch configurations near point-vortex equilibria by introducing a centralized center condition and an adjustable background strength. Using a nonlinear implicit-function theorem together with a bootstrap argument, the authors prove the existence of -regular, convex patch boundaries and extend the construction to general -patch lattices. The work provides a rigorous route to patch lattices in a compact, translationally-invariant setting and enhances understanding of near-equilibrium vortex structures in 2D turbulence. Key techniques include contour dynamics, Green-function analysis on the torus, Kirchhoff-Routh functionals, and a careful linearization/regularity bootstrap strategy.

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 patches near a point vortex equilibrium. Different with the case of bounded domains in , a constant background vorticity will arise from the compact nature of flat torus, and the -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 regularity and convexity of boundary curves.
Paper Structure (14 sections, 15 theorems, 157 equations)

This paper contains 14 sections, 15 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Mathematical formulation and main results
  3. The Green function and contour dynamics
  4. Expansion for the Green function $G$
  5. Equilibrium states for point vortices in $\mathbb{T}^2$
  6. Contour dynamics for single-layered patches
  7. The functional analysis frame for single-layered patches
  8. Regularity of the contour dynamic equation
  9. Linearization at $F (0, \gamma,1)$
  10. The existence for single-layered patches and other properties
  11. The existence by implicit function theorem
  12. The regularity and convexity
  13. Construction for the general case
  14. The $P$-function and its logarithmic derivative

Key Result

Theorem 2.1

There exists $\varepsilon_0>0$ such that for any $\varepsilon\in [0,\varepsilon_0)$, 1-1 has a steady patch solution 1-3, where the strength of background vorticity $\gamma_\varepsilon$ satisfies

Theorems & Definitions (29)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 4.1
  • proof
  • ...and 19 more