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An Onsager-type theorem for SQG

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arxiv 2407.02582 v1 pith:C72CS3VQ submitted 2024-07-02 math.AP

Mimi Dai , Vikram Giri , Razvan-Octavian Radu This is my paper · ORCID
classification math.AP
keywords conservationsolutionsweakachievedclasscompactconjectureconstruct
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abstract

We construct non-trivial weak solutions $\theta\in C_t^0C_x^{0-}$ 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-Vicol, 2015) and resolves the Onsager conjecture for SQG. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced in Giri-Radu (2023).

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.AP 2025-02 conditional novelty 8.0 of 10

    Non-uniqueness with forcing for alpha-SQG is established across the full supercritical Sobolev range s < alpha + 2/p, using new smooth compactly supported unstable vortices.

  2. Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation

    math.AP 2025-09 accept novelty 7.0 of 10

    For any L^{4/3}_x initial datum there exists a global weak solution of SQG conserving the H^{-1/2}_x Hamiltonian, obtained via a vanishing-viscosity limit with no anomalous dissipation.

  3. Strong ill-posedness and non-existence in Sobolev spaces for generalized-SQG

    math.AP 2025-02 accept novelty 6.0 of 10

    For generalized SQG with gamma in (-1,1), the paper proves strong ill-posedness and instantaneous non-existence in H^beta for beta in [1,2+gamma) intersect (3/2+gamma,2+gamma).

  4. An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions

    math.AP 2025-02 conditional novelty 6.0 of 10

    For every γ<1/3, there exist compactly supported weak solutions (v,θ) to the 2D Euler-Boussinesq system in C^γ(R×T^2) × C^γ(R×T^2) that violate conservation of the temperature's L^p-norm.

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