Logarithmic estimates for continuity equations

The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in can be used to provide an alternative proof of the result in, where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions.