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Existence and Uniqueness for the SQG Vortex-Wave System when the Vorticity is Constant near the Point-Vortex

Dimitri Cobb, Martin Donati, Ludovic Godard-Cadillac

TL;DR

This work analyzes the vortex-wave system for the SQG equation with $0<s<1$, establishing local well-posedness in $H^4$ under a plateau condition near the point vortex and proving $H^2$-stability along with a blow-up criterion. It then treats global weak solutions in the sub-critical regime $s>1/2$ and develops a $V$-weak notion to obtain global existence at the critical case $s=1/2$, using commutator structures and kernel regularization. The authors build a robust functional-analytic framework employing Besov spaces and Fourier analysis, derive precise a priori estimates and convergence results for regularized problems, and extend the results to multiple vortices under non-collapsing assumptions. The results significantly advance understanding of SQG vortex–wave dynamics, providing rigorous existence, stability, and blow-up criteria in regimes where velocity fields are singular and standard transport theory is inapplicable, with potential implications for geophysical flow modeling and singularity formation analysis.

Abstract

This article studies the vortex-wave system for the Surface Quasi-Geostrophic equation with parameter 0 < s < 1. We obtained local existence of classical solutions in H^4 under the standard ''plateau hypothesis'', H^2-stability of the solutions, and a blow-up criterion. In the sub-critical case s > 1/2 we established global existence of weak solutions. For the critical case s = 1/2, we introduced a weaker notion of solution (V-weak solutions) to give a meaning to the equation and prove global existence.

Existence and Uniqueness for the SQG Vortex-Wave System when the Vorticity is Constant near the Point-Vortex

TL;DR

This work analyzes the vortex-wave system for the SQG equation with , establishing local well-posedness in under a plateau condition near the point vortex and proving -stability along with a blow-up criterion. It then treats global weak solutions in the sub-critical regime and develops a -weak notion to obtain global existence at the critical case , using commutator structures and kernel regularization. The authors build a robust functional-analytic framework employing Besov spaces and Fourier analysis, derive precise a priori estimates and convergence results for regularized problems, and extend the results to multiple vortices under non-collapsing assumptions. The results significantly advance understanding of SQG vortex–wave dynamics, providing rigorous existence, stability, and blow-up criteria in regimes where velocity fields are singular and standard transport theory is inapplicable, with potential implications for geophysical flow modeling and singularity formation analysis.

Abstract

This article studies the vortex-wave system for the Surface Quasi-Geostrophic equation with parameter 0 < s < 1. We obtained local existence of classical solutions in H^4 under the standard ''plateau hypothesis'', H^2-stability of the solutions, and a blow-up criterion. In the sub-critical case s > 1/2 we established global existence of weak solutions. For the critical case s = 1/2, we introduced a weaker notion of solution (V-weak solutions) to give a meaning to the equation and prove global existence.
Paper Structure (50 sections, 35 theorems, 278 equations)

This paper contains 50 sections, 35 theorems, 278 equations.

Table of Contents

  1. Presentation of the Problem and Main Result
  2. The Vortex-Wave System for SQG
  3. Singular vorticity: The point-vortex system
  4. Local Existence and Uniqueness for the SQG Vortex-Wave
  5. Global existence of weak solution in the sub-critical case
  6. Global Existence of Weak Solutions in the Critical Case
  7. Duhamel Formula for the Quasi-Point-Vortex System
  8. A Remark on the Structure of the Non-Linearity
  9. A Weaker Notion of Weak Solution
  10. Global Existence of $V$-Weak Solutions in the Critical Case
  11. Functional Analysis and Regularization
  12. Fourier Analysis and Besov spaces
  13. Commutator formulation
  14. Regularization of the Kernel
  15. Local Existence of Classical Solutions
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $0 < s < 1$. Consider $\theta_0 \in H^k$ an initial datum for some integer $k \geq 4$ and $z_0 \in \mathbb{R}^2$ an initial point-vortex position. Assume that $\theta_0$ is constant on some neighborhood of $z_0$ (plateau assumption). Then the following assertions hold true.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Global weak solutions if $s>1/2$
  • Lemma 1.4: Commutator formulation
  • Definition 1.5: $V$-weak solutions
  • Definition 1.6: $\Gamma$-coincidence
  • Theorem 1.8: Global $V$-weak solutions to the quasi vortex-wave if $s=1/2$
  • Definition 2.1: Littlewood-Paley decomposition
  • Definition 2.2: Inhomogeneous Besov spaces
  • Definition 2.3: Homogeneous Besov spaces
  • ...and 54 more