This talk will discuss the properties of solutions to linear advection
or continuity equations, or equivalent to flows of ODE's, with
non-smooth velocity fields that belong to some Sobolev space.
This type of results is also connected to the existence of strong solutions (or pathwise uniqueness) to SDE's.
We will start with the theory of renormalized solutions which provides 0qualitative arguments of regularity and well-posedness and move to more recent quantitative approaches.
Those estimates can be used to control the mixing properties of the flow but are also critical for many applications to non-linear systems.