Classical flows of vector fields with exponential or sub-exponential summability

We show that vector fields b whose spatial derivative Dx b satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if Dx b satisfies a suitable exponential summability condition then the flow associated to b has Sobolev regularity, without assuming boundedness of divx b. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations.