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Peer-reviewedCC BYJournal of the European Mathematical Society

An Onsager-type theorem for SQG equations

  • Mimi Dai1
  • Vikram Giri2
  • Răzvan-Octavian Radu3
  1. 1University of Illinois Chicago, USA
  2. 2ETH Zürich, Switzerland
  3. 3Princeton University, USA
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Abstract

We construct non-trivial weak solutions \theta\in C_{t}^{0}C_{x}^{0-} to the surface quasi-geostrophic (SQG) equations, which have compact support in time and thus violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class C_{t,x}^{0} \subset L_{t,x}^{3} (Isett and Vicol (2015)) and resolves the Onsager conjecture for SQG equations. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced by Giri and Radu (2023).

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Cite

BibTeXRIS
BibTeX
@article{dai2026onsagertype,
  title = {{An Onsager-type theorem for SQG equations}},
  author = {Mimi Dai and Vikram Giri and Răzvan-Octavian Radu},
  journal = {Journal of the European Mathematical Society},
  year = {2026},
  month = oct,
  doi = {10.4171/jems/1812},
  url = {https://doi.org/10.4171/jems/1812},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - An Onsager-type theorem for SQG equations
AU  - Mimi Dai
AU  - Vikram Giri
AU  - Răzvan-Octavian Radu
PY  - 2026
DA  - 2026/10/21
JO  - Journal of the European Mathematical Society
DO  - 10.4171/jems/1812
UR  - https://doi.org/10.4171/jems/1812
AB  - We construct non-trivial weak solutions \theta\in C_{t}^{0}C_{x}^{0-} to the surface quasi-geostrophic (SQG) equations, which have compact support in time and thus violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class C_{t,x}^{0} \subset L_{t,x}^{3} (Isett and Vicol (2015)) and resolves the Onsager conjecture for SQG equations. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced by Giri and Radu (2023).
SN  - 1435-9855
SN  - 1435-9863
ER  -