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arxiv: 2403.08279 · v1 · pith:5K2ECQIR · submitted 2024-03-13 · math.AP

On the conservation laws and the structure of the nonlinearity for SQG and its generalizations

Philip Isett , Andrew Ma This is my paper

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keywords nonlinearityangularconservationmomentummsqgoptimalprovesolutions
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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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  1. Hamiltonian compactness and dissipation for the generalized SQG equation in the inviscid limit

    math.AP 2026-04 unverdicted novelty 7.0

    Strong compactness in the lowest norm making the nonlinearity well-defined prevents anomalous Hamiltonian dissipation for the generalized SQG equation in the inviscid limit, yielding global conservative weak solutions...