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Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

Ángel Castro, Daniel Faraco, Francisco Mengual, Marcos Solera

TL;DR

This work proves non-uniqueness of forced, weak solutions to the generalized SQG equation in the supercritical Sobolev regime $s<α+\frac{2}{p}$ for all $0\le α\le1$, extending Vishik’s instability framework to the α-SQG family. The authors construct smooth, compactly supported vortices that are linearly and nonlinearly unstable in self-similar coordinates, and then leverage a nonlinear bootstrap to generate uncountably many forced solutions from zero initial data. A by-product is the existence of global smooth solutions to the unforced equation that are neither rotating nor traveling, obtained by reversing the instability flow. They also establish a Hamiltonian identity and a renormalization property for these solutions, situating the results within the broader context of conservation laws and Onsager-type conjectures for active scalars. The techniques hinge on a careful analysis of the self-similar stability operator, a piecewise-constant vortex construction, and a nuanced treatment of the nonlocal Biot–Savart law, including a Golovkin-type argument to connect linear instability to non-uniqueness.

Abstract

We prove non-uniqueness of weak solutions to the forced $α$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < α+ \frac{2}{p}$, covering the 2D Euler equation ($α= 0$), the Surface Quasi-Geostrophic equation ($α= 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $α$-SQG equation that are neither rotating nor traveling.

Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

TL;DR

This work proves non-uniqueness of forced, weak solutions to the generalized SQG equation in the supercritical Sobolev regime for all , extending Vishik’s instability framework to the α-SQG family. The authors construct smooth, compactly supported vortices that are linearly and nonlinearly unstable in self-similar coordinates, and then leverage a nonlinear bootstrap to generate uncountably many forced solutions from zero initial data. A by-product is the existence of global smooth solutions to the unforced equation that are neither rotating nor traveling, obtained by reversing the instability flow. They also establish a Hamiltonian identity and a renormalization property for these solutions, situating the results within the broader context of conservation laws and Onsager-type conjectures for active scalars. The techniques hinge on a careful analysis of the self-similar stability operator, a piecewise-constant vortex construction, and a nuanced treatment of the nonlocal Biot–Savart law, including a Golovkin-type argument to connect linear instability to non-uniqueness.

Abstract

We prove non-uniqueness of weak solutions to the forced -SQG equation with Sobolev regularity in the supercritical regime , covering the 2D Euler equation (), the Surface Quasi-Geostrophic equation (), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) -SQG equation that are neither rotating nor traveling.

Paper Structure

This paper contains 57 sections, 83 theorems, 547 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Improvements and corollaries
  3. Main novelties
  4. Unstable vortices
  5. Global smooth solutions
  6. Hamiltonian Identity
  7. Renormalization property
  8. Organization of the paper
  9. Stability equation in self-similar coordinates
  10. Self-similar coordinates
  11. Linear stability equation
  12. Vortices
  13. Purely $n$-fold symmetric temperatures
  14. Non-linear stability equation
  15. Space of $n$-fold symmetric temperatures
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let $0\leq\alpha\leq 1$, $s\geq 0$ and $1\leq p\leq\infty$ satisfying There exist $T>0$ and a force forceintegrability such that there are uncountably many solutions to the $\alpha$-SQG equation eq:SQG starting from $\theta^\circ=0$.

Figures (1)

  • Figure 1: Representation of the Sobolev spaces $W^{s,p}$ in terms of the variables $1/p$ and $s$. The green region corresponds to the well-posedness regime $s>\alpha+2/p$. The black segment corresponds to the critical line $s=\alpha+2/p$. The light red region corresponds to the (expected) non-uniqueness regime $s<\alpha+2/p$. The red area $s\leq\alpha+2/p-\varepsilon$ corresponds to the non-uniqueness with forcing, as established in Theorem \ref{['thm:main']}, where $\varepsilon>0$ is arbitrarily small. The red segment corresponds to Vishik's non-uniqueness theorem ($\alpha=s=0$). In this context, the green point $(1/p,s)=(0,0)$ corresponds to the Yudovich well-posedness class.

Theorems & Definitions (156)

  • Theorem 1.1: Sobolev non-uniqueness
  • Corollary 1.1: Hölder non-uniqueness
  • Theorem 1.2: Non-uniqueness at $t=-\infty$
  • Corollary 1.2: Non-linearly unstable vortex
  • Corollary 1.3: Global smooth solutions
  • Corollary 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Remark 2.1
  • ...and 146 more