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Peer-reviewedCC BYArchive for Rational Mechanics and Analysis

Unstable Vortices, Sharp Non-Uniqueness with Forcing, and Global Smooth Solutions for the SQG Equation

  • Ángel Castro1
  • Daniel Faraco2
  • Francisco Mengual3
  • Marcos Solera4
4 affiliations
  1. 1Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain
  2. 2Departamento de Matemáticas, Universidad Autónoma de Madrid, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain
  3. 3Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
  4. 4Departament d'Anàlisi Matemàtica, Universitat de València, 46100 Burjassot, Spain
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Abstract

We prove that non-uniqueness of weak solutions to the forced α α -SQG equation with Sobolev regularity Ws,p W s , p in the supercritical regime s < α + 2/p s < α + 2 p , covering the 2D Euler equation ( α = 0 α = 0 ), the Surface Quasi-Geostrophic equation ( α = 1 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 that existence of global smooth solutions to the (unforced) α α -SQG equation that are neither rotating nor traveling.

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Cite

BibTeXRIS
BibTeX
@article{castro2026unstable,
  title = {{Unstable Vortices, Sharp Non-Uniqueness with Forcing, and Global Smooth Solutions for the SQG Equation}},
  author = {Ángel Castro and Daniel Faraco and Francisco Mengual and Marcos Solera},
  journal = {Archive for Rational Mechanics and Analysis},
  year = {2026},
  month = oct,
  doi = {10.1007/s00205-026-02252-x},
  url = {https://doi.org/10.1007/s00205-026-02252-x},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Unstable Vortices, Sharp Non-Uniqueness with Forcing, and Global Smooth Solutions for the SQG Equation
AU  - Ángel Castro
AU  - Daniel Faraco
AU  - Francisco Mengual
AU  - Marcos Solera
PY  - 2026
DA  - 2026/10/03
JO  - Archive for Rational Mechanics and Analysis
DO  - 10.1007/s00205-026-02252-x
UR  - https://doi.org/10.1007/s00205-026-02252-x
AB  - We prove that non-uniqueness of weak solutions to the forced α α -SQG equation with Sobolev regularity W^s,p W s , p in the supercritical regime s < α + 2/p s < α + 2 p , covering the 2D Euler equation ( α = 0 α = 0 ), the Surface Quasi-Geostrophic equation ( α = 1 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 that existence of global smooth solutions to the (unforced) α α -SQG equation that are neither rotating nor traveling.
SN  - 0003-9527
SN  - 1432-0673
ER  -