The flow associated to weakly differentiable vector fields: Recent results and open problems

Abstract

In this note we describe some recent developments of the theory of flows associated to vector fields with a low regularity with respect to the spatial variables, for instance with a Sobolev or BV regularity. After the illustration of some applications of this theory to conservation laws and PDE's in fluid dynamics, we give an axiomatic presentation of the problem, based on a probabilistic approach inspired by the work of L.C. Young. In the final part we discuss very recent results on the regularity of the flow itself with respect to the spatial variables.