  We show that vector fields $b$ whose spatial derivative $D_xb$ 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
$D_xb$ satisfies a suitable exponential summability condition then the flow
associated to $b$ has Sobolev regularity, without assuming boundedness of ${\rm
div}_xb$. We then apply these results to the representation and Sobolev
regularity of weak solutions of the Cauchy problem for the transport and
continuity equations.
