About Equations with Drift and Diffusion

We will discuss the parabolic equation which consists on the heat equation plus a first order term. This term is a vector field (drift) multiplied by the gradient of the solution. We can interpret this equation as representing the evolution of temperature in a moving fluid. The question we explore is what hypothesis the vector field must satisfy so that the solution to the equation does not develop a discontinuity in finite time. We will focus especially in the case of drifts which are of divergence zero, given their applications to problems in incompressible fluids.