On some typicality and density results for nonsmooth vector fields and the associated continuity equation.

We are interested in the problem of existence and/or uniqueness of local distributional solutions to the continuity equation associated with a vector field in $L^1_tL^q_x$ where $q\in [1,\infty)$. It is well known that existence of $L^{\infty}_tL^{q’}_x$ solutions holds for any initial datum in $L^{q’}$ provided the vector field has bounded divergence. Moreover, DiPerna and Lions' theory guarantees uniqueness of such solutions when the vector field belongs to $L^1_tW^{1,q}_x$. Unfortunately, in $L^1_tL^q_x$ only “a few” vector fields satisfy these conditions. Therefore a natural question is: how “typical” is it in $L^1_tL^q_x$ for the existence and/or uniqueness properties to hold without making any further assumptions on the vector field? In this talk we will address this question and we will provide an answer in terms of Baire category. This is a joint work with S. Modena (GSSI).