Osgood meets DiPerna-Lions

This thesis investigates the Cauchy problem for the transport equation driven by vector fields with low regularity. We begin by presenting the classical theory developed by DiPerna and Lions, which establishes well-posedness under the assumption that the vector field belongs to a suitable Sobolev space. In the second part, we extend these results to a broader class of vector fields that do not necessarily have integrable derivatives but still satisfy a weaker form of regularity. These results contribute to the ongoing effort to understand the minimal regularity required to ensure well-posedness for linear transport equations.