  We consider a number of uniqueness questions for several wide classes of
active scalar equations, unifying and generalizing the techniques of several
authors. As special cases of our results, we provide a significantly simplified
proof to the known uniqueness result for the 2D Euler equations in $L^1 \cap
BMO$ and provide a mild improvement to the recent results of Rusin for the 2D
inviscid surface quasi-geostrophic (SQG) equations, which are now to our
knowledge, the best results known for this model. We also obtain what are (to
our knowledge) the strongest known uniqueness results for the
Patlak-Keller-Segel models. We obtain these results via technical refinements
of energy methods which are well-known in the $L^2$ setting but are less
well-known in the $\dot{H}^{-1}$ setting. The $\dot{H}^{-1}$ method can be
considered a generalization of Yudovich's classical method and is naturally
applied to equations such as the Patlak-Keller-Segel models with nonlinear
diffusion, and other variants. Important points of our analysis are an
$L^p$-$BMO$ interpolation lemma and a Sobolev embedding lemma which shows that
velocity fields $v$ with $\grad v \in BMO$ are locally log-Lipschitz; the
latter is known in harmonic analysis but does not seem to have been connected
to this setting.
