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On the conservation laws and the structure of the nonlinearity for SQG and its generalizations
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Using a new definition for the nonlinear term, we prove that all weak solutions to the SQG equation (and mSQG) conserve the angular momentum. This result is new for the weak solutions of [Resnick, '95] and rules out the possibility of anomalous dissipation of angular momentum. We also prove conservation of the Hamiltonian under conjecturally optimal assumptions, sharpening a well-known criterion of [Cheskidov-Constantin-Friedlander-Shvydkoy, '08]. Moreover, we show that our new estimate for the nonlinearity is optimal and that it characterizes the mSQG nonlinearity uniquely among active scalar nonlinearities with a scaling symmetry.
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Cited by 3 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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An Onsager-type Theorem for General 2D Active Scalar Equations
For odd, homogeneous, order-delta multipliers with -1 <= delta <= 0, there exist non-trivial weak solutions failing to conserve the Hamiltonian at every regularity Lambda^{-1}theta in C^gamma with gamma < 1 + 2delta/3...
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Weak Solutions for Inviscid SQG with Lorentz Data
Inviscid SQG admits global Hamiltonian-conserving weak solutions for every initial datum in the critical Lorentz space L^{4/3,2} on R^2 and smooth bounded domains.
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