Extra regularity for parabolic equations with drift terms

We consider a parabolic equation with a drift term u+bu–u_t=0. Under some natural conditions on the vector valued function b, we prove that solutions possess extra regularity and better qualitative behavior than those provided by standard theory. For example, we show that the fundamental solution has global Gaussian upper bound even for some b with a large singularity in the form of c/|x|. We also show that bounded solutions are Hölder continuous when |b|² is just form bounded and divergence free, a case where not even continuity is expected. A Liouville type theorem is also proven.