Transport equations arise in various areas
of fluid mechanics, but the precise conditions on the vector field for
them to be well-posed are still not fully understood.
The renormalized theory of DiPerna and Lions
for linear transport equations with unsmooth coefficient uses the tools
of approximation of an arbitrary weak solution by smooth functions,
and the renormalization property, that is to say to write down an
equation on a nonlinear function of the solution. Under some $W^{1,1}$
regularity assumption on the coefficient, well-posedness holds.
In this paper, we establish that these properties are indeed equivalent
to the uniqueness of weak solutions to the Cauchy problem, without
any regularity assumption on the coefficient. Coefficients with
unbounded divergence but with bounded compression are also considered.

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