I am curious about weak solutions of the parabolic problem 
$ u\_t - \Delta u + a(x,t) \cdot \nabla u = f(x,t)$ in $ \Omega \times (0,\infty)$ with $ u=0$ on $ \partial \Omega$. Here $a(x,t)$ is some divergence free vector field with minimal smoothness properties. We assume that $ u(x,0)$ is smooth. 


My question is: what minimal properties can I assume of $a(x,t)$ which will imply
that $u(x,t)$ is locally bounded in time? 


In the elliptic case one essentially doesn't need any restrictions on $a$ besides the diverge free property. I suspect this is the case for the parabolic versions also but I am curious.  

Lately these questions are a fairly hot topic but there people want to put minimal conditions on $a$ so as to have the solutions Holder continuous. 


thanks
Craig



Regularity of parabolic equation; divergence free drift
The $L^\infty$ a priori estimate for u does not depend on $a$ at all provided that the initial data is either bounded (you would apply maximum principle) or $L^1$ (you would apply Nash bound on the heat kernel).


The only subtlety (if any) would be at the definition of weak solution. If $a$ is too rough, weak solutions may not be unique and then everything breaks. There is a series of papers by Pierre Louis Lions and Claude Le Bris discussing the issue of uniqueness.


