  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.
