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Global well-posedness for supercritical SQG with perturbations of radially symmetric data

Aynur Bulut, Hongjie Dong

Abstract

We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.

Global well-posedness for supercritical SQG with perturbations of radially symmetric data

Abstract

We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.
Paper Structure (3 sections, 2 theorems, 35 equations)

This paper contains 3 sections, 2 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Motivating decomposition: perturbation lemma
  3. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Fix $0<\gamma<1$ and set $s=2-\gamma$. Then there exists $\epsilon>0$ such that if $f\in L^1(\mathbb{R}^2)\cap H^s(\mathbb{R}^2)$ is radially symmetric with $f\geq 0$, and $g\in H^{2}(\mathbb{R}^2)$ satisfies then the initial value problem has a unique global solution in $C([0,\infty);H^{2-\gamma}(\mathbb{R}^2))$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm1']}