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Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

Junekey Jeon, Andrej Zlatos

TL;DR

The paper develops a local well-posedness theory for the generalized SQG equation in low-regularity settings by employing a Lagrangian formulation grounded in generalized layer cake representations. It proves local existence, uniqueness, and flow-map stability in spaces $L^1\cap L^\infty$ with a modulus of continuity, controlled by a Lipschitz bound $L_{\mu}(\Theta)$. Furthermore, it shows that if level-set boundaries are $H^2$, the evolution preserves a structured geometric regularity for a nontrivial time, with explicit blow-up criteria tied to the growth of $Q(\Theta)$ and $L_{\mu}(\Theta)$; for $\alpha\le 1/6$, the authors prove that finite-time breakdown must correspond to loss of $H^2$-regularity of level sets rather than level-set collisions or pile-ups, providing a sharp geometric mechanism for singularity formation.

Abstract

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially Hölder continuous with Hölder exponents depending on the equation parameter $α\in(0,\frac 12)$) that have $H^2$ level sets (i.e., with $L^2$ curvatures). Moreover, for $α\le\frac 16$ and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose $H^2$-regularity, and hence not just due to level set collisions or "pile ups".

Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

TL;DR

The paper develops a local well-posedness theory for the generalized SQG equation in low-regularity settings by employing a Lagrangian formulation grounded in generalized layer cake representations. It proves local existence, uniqueness, and flow-map stability in spaces with a modulus of continuity, controlled by a Lipschitz bound . Furthermore, it shows that if level-set boundaries are , the evolution preserves a structured geometric regularity for a nontrivial time, with explicit blow-up criteria tied to the growth of and ; for , the authors prove that finite-time breakdown must correspond to loss of -regularity of level sets rather than level-set collisions or pile-ups, providing a sharp geometric mechanism for singularity formation.

Abstract

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially Hölder continuous with Hölder exponents depending on the equation parameter ) that have level sets (i.e., with curvatures). Moreover, for and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose -regularity, and hence not just due to level set collisions or "pile ups".

Paper Structure

This paper contains 4 sections, 21 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['T1.2']}
  3. Proof of Theorem \ref{['T1.4']}(i)--(ii)
  4. Proof of Theorem \ref{['T1.4']}(iii)

Key Result

Theorem 1.3

Assume that $\theta^{0}\in L^{1}({\mathbb{R}}^{2})\cap L^{\infty}({\mathbb{R}}^{2})$ admits a generalized layer cake representation $(\Theta,\mu)$ with $L_{\mu}(\Theta)<\infty$. Then there is an open interval $I\ni 0$ and a Lagrangian solution $\theta$ to 1.1--1.2 on $I$ with $\alpha\in(0,\frac{1}{2

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 32 more