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

We show that vector fields b whose spatial derivative D_x 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 D_x b satisfies a suitable exponential summability condition then the flow associated to b has Sobolev regularity, without assuming boundedness of div_x 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.