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A proof of Onsager's Conjecture for the SQG equation
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We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG.
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Cited by 4 Pith papers
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Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation
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.
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Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
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.
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Strong ill-posedness and non-existence in Sobolev spaces for generalized-SQG
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).
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An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions
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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