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.