  We consider transport equations with an incompressible transporting vector
field. Whereas smooth solutions of such equations conserve every $L^p$ norm
simply by the chain rule, the question arises how regular a weak solution needs
to be to guarantee this conservation property. Whereas the classical
DiPerna-Lions theory gives sufficient conditions in terms of the regularity of
the coefficients, with no regularity requirement on the transported scalar, we
give here sufficient conditions in terms of the combined regularities of the
coefficients and the scalar. This is motivated by the case of active scalar
equations, where the transporting vector field has the same regularity as the
transported scalar. We use commutator estimates similar to those of
Constantin-E-Titi in the context of Onsager's conjecture, but we require novel
arguments to handle the case of $L^p$ norms when $p\neq 2$.
