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, matching the rigid bound.
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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, matching the rigid bound.