Since the work by DiPerna and Lions (89) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. In this talk, we give an overview of this theory presenting classical results and new quantitative estimates. One important tool in our investigation is a Kakeya type singular operator. We establish the weak type (1,1) bound for this operator and we exploit it to prove well-posedness and stability results for the continuity and transport equation associated to vector fields represented as singular integrals of BV functions. We also discuss the optimality of this result.

Finally, we present sharp regularity estimates for solutions of the continuity equation under various assumptions on the velocity fields.