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Existence of nested polygonal vortex patches for the generalized SQG equation

Edison Cuba, Lucas C. F. Ferreira

TL;DR

The paper addresses the construction of time-periodic, co-rotating nested polygon vortex patches for the generalized SQG equations in the singular regime $α ∈ [1,2)$ and, in the SQG limit $α=1$, on the full plane. The authors combine contour-dynamics desingularization with a carefully tailored implicit-function theorem in Banach spaces to perturb from a nondegenerate point-vortex equilibrium and produce a $C^1$ curve of solutions consisting of $2m+1$ nested polygons with precise boundary geometry. They establish the necessary regularity of the nonlinear boundary-functional $ ext{F}^α$ and show its linearization is an isomorphism under a determinant nondegeneracy condition, enabling the existence and uniqueness of the perturbed solutions. The main contributions include rigorous existence results for highly symmetric nested polygonal V-states in the gSQG range, explicit asymptotic expansions for boundary profiles, and convexity of the resulting patches, thereby extending prior work to multi-layer nested configurations in the highly singular regime. These results provide a constructive, geometrically explicit framework for coherent rotating structures in active scalar equations with singular velocity coupling, with potential implications for geophysical fluid dynamics and related vortex-patch analyses.

Abstract

This paper investigates time-periodic solutions of both the surface quasi-geostrophic (SQG) equation and its generalized form (gSQG) within the more singular regime, focusing on the evolution of patch-type structures. Assuming the underlying point vortex equilibrium satisfies a natural nondegeneracy condition, we employ an implicit function argument to construct families of co-rotating nested polygonal vortex patch solutions. These configurations provide precise asymptotic descriptions of the geometry of the evolving patch boundaries. Our results contribute to the broader understanding of coherent rotating structures arising in active scalar equations with singular velocity coupling.

Existence of nested polygonal vortex patches for the generalized SQG equation

TL;DR

The paper addresses the construction of time-periodic, co-rotating nested polygon vortex patches for the generalized SQG equations in the singular regime and, in the SQG limit , on the full plane. The authors combine contour-dynamics desingularization with a carefully tailored implicit-function theorem in Banach spaces to perturb from a nondegenerate point-vortex equilibrium and produce a curve of solutions consisting of nested polygons with precise boundary geometry. They establish the necessary regularity of the nonlinear boundary-functional and show its linearization is an isomorphism under a determinant nondegeneracy condition, enabling the existence and uniqueness of the perturbed solutions. The main contributions include rigorous existence results for highly symmetric nested polygonal V-states in the gSQG range, explicit asymptotic expansions for boundary profiles, and convexity of the resulting patches, thereby extending prior work to multi-layer nested configurations in the highly singular regime. These results provide a constructive, geometrically explicit framework for coherent rotating structures in active scalar equations with singular velocity coupling, with potential implications for geophysical fluid dynamics and related vortex-patch analyses.

Abstract

This paper investigates time-periodic solutions of both the surface quasi-geostrophic (SQG) equation and its generalized form (gSQG) within the more singular regime, focusing on the evolution of patch-type structures. Assuming the underlying point vortex equilibrium satisfies a natural nondegeneracy condition, we employ an implicit function argument to construct families of co-rotating nested polygonal vortex patch solutions. These configurations provide precise asymptotic descriptions of the geometry of the evolving patch boundaries. Our results contribute to the broader understanding of coherent rotating structures arising in active scalar equations with singular velocity coupling.

Paper Structure

This paper contains 7 sections, 13 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Boundary equations
  3. Co-rotating nested polygons patches
  4. Notation and functional spaces
  5. Regularity and linearization analysis on the functional
  6. Existence of nested polygons for gSQG equation
  7. Existence of nested polygons for SQG equation

Key Result

Theorem 1.1

Let $\alpha \in [1,2)$, $\vartheta \in \{0,1\}$, $b_1, b_2 \in (0,1)$, $\gamma_0, \gamma_1 \in \mathbb{R} \setminus \{0\}$, and $d_1, d_2 \in (0,\infty)$. Suppose that the pair $(\Omega^*, \gamma_2^*) \in (\mathbb{R} \setminus \{0\})^2$ solves system syst-pj2 and satisfies the non-degeneracy conditi with the sets $\mathcal{D}_{jk}^\varepsilon$ defined by produces a uniformly rotating solution to

Theorems & Definitions (26)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.5
  • ...and 16 more