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Doubly Connected V-States in Geophysical Models: A General Framework

Taoufik Hmidi, Liutang Xue, Zhilong Xue

TL;DR

This work establishes the existence of doubly connected V-states (rotating patches) near annuli for a broad class of active scalar equations with kernels that decompose as $K_0(|\mathbf{x}-\mathbf{y}|)+K_1(\mathbf{x},\mathbf{y})$ and satisfy complete monotonicity, integrability, and symmetry assumptions. The authors develop a universal spectral framework using functions $φ_n$ and $φ_{n,b}$ to factorize the spectrum into a pair of branches $Ω^{±}_{n,b}$, and apply Crandall–Rabinowitz bifurcation theory to obtain two $m$-fold symmetric families of V-states bifurcating from annuli, for sufficiently large $m$ and suitable $b$. They provide detailed spectral analysis, positivity/monotonicity properties, and real-analytic dependence on parameters, unifying and extending known results for 2D Euler, gSQG, and QGSW in both whole-space and bounded-domain settings, including disc and annulus. The framework yields new radial/annular results and explicit conditions (or regimes) ensuring discriminant positivity, thereby producing bifurcating branches across diverse geophysical flow models with rigorous control on regularity and stability features.

Abstract

In this paper, we prove the existence of doubly connected V-states (rotating patches) close to an annulus for active scalar equations with completely monotone kernels. This provides a unified framework for various results related to geophysical flows. This allows us to recover existing results on this topic while also extending to new models, such as the gSQG and QGSW equations in radial domains and 2D Euler equation in annular domains.

Doubly Connected V-States in Geophysical Models: A General Framework

TL;DR

This work establishes the existence of doubly connected V-states (rotating patches) near annuli for a broad class of active scalar equations with kernels that decompose as and satisfy complete monotonicity, integrability, and symmetry assumptions. The authors develop a universal spectral framework using functions and to factorize the spectrum into a pair of branches , and apply Crandall–Rabinowitz bifurcation theory to obtain two -fold symmetric families of V-states bifurcating from annuli, for sufficiently large and suitable . They provide detailed spectral analysis, positivity/monotonicity properties, and real-analytic dependence on parameters, unifying and extending known results for 2D Euler, gSQG, and QGSW in both whole-space and bounded-domain settings, including disc and annulus. The framework yields new radial/annular results and explicit conditions (or regimes) ensuring discriminant positivity, thereby producing bifurcating branches across diverse geophysical flow models with rigorous control on regularity and stability features.

Abstract

In this paper, we prove the existence of doubly connected V-states (rotating patches) close to an annulus for active scalar equations with completely monotone kernels. This provides a unified framework for various results related to geophysical flows. This allows us to recover existing results on this topic while also extending to new models, such as the gSQG and QGSW equations in radial domains and 2D Euler equation in annular domains.

Paper Structure

This paper contains 26 sections, 24 theorems, 381 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Boundary equation of doubly connected V-states and linearization
  3. Boundary equation
  4. Linearization
  5. Spectrum
  6. Analysis of the universal functions
  7. Positivity and Monotonicity of $\phi_{n,b}(x)$
  8. Analyticity of universal functions and related spectral terms
  9. Spectral Study
  10. Pointwise bounds of several spectral terms
  11. Asymptotic structure of the discriminant
  12. Spectrum distribution
  13. Proof of main theorem
  14. Applications to geophysical flows
  15. 2D Euler equation in the whole space
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose that $K(\mathbf{x},\mathbf{y})=K_0(|\mathbf{x}-\mathbf{y}|)+K_1(\mathbf{x},\mathbf{y})$ meets the conditions A1-A4. Set Then, for any $b\in \mathcal{S}$, there exists a sufficiently large number $m_0\in \mathbb{N}^\star$, such that for any $m\geqslant m_0$, the active scalar equation eq:ASE-eq:stream_function admits two families of $m$-fold symmetric doubly connected V-states with $C^{2-

Figures (5)

  • Figure 1: The graph of $x\mapsto \Psi_b(x)$ for $b=0.5$.
  • Figure 2: The graph of $x\mapsto \Psi_b(x)$ for different values of $b$.
  • Figure 3: The graph of $x\mapsto b\phi_{1,b}(x)$ for different values of $b$.
  • Figure 4: The graph of $\color[rgb]{0,0.447,0.741}{x\mapsto \frac{1}{b}\phi_{1,b}(x)-\phi_1(bx)}$ and $\color[rgb]{0.85,0.325,0.098}{x\mapsto \Psi_b(x)}$ for $b=0.1$.
  • Figure 5: The contour $C_{h,l,r}$.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lem:phi_n_b_1']}
  • Lemma 2
  • Remark 2
  • proof : Proof of Lemma \ref{['lem:phi-n-b-mono']}
  • Lemma 3
  • ...and 39 more