What's the meaning of a continuity equation with $\nabla^2 \rho$ on the right-hand side? I stumbled upon a continuity equation with a $\nabla^2$ term on the right-hand side:
$$  \partial_t \rho + \nabla (\vec b \rho) = D \nabla^2 \rho ,  $$ 
where $b$ denotes the forward velocity and $D$ is a constant. 
What's the meaning of such a diffusion equation?

Some background: 
Since, we have particle number conservation, we have
$$  \partial_t \rho + \nabla (\vec v \rho) = 0  , $$ 
where $v$ denotes the ordinary flux velocity. Moreover, if there are sources, we have
$$  \partial_t \rho + \nabla (\vec v \rho) = \sigma . $$ 
 A: It is the so called  convection–diffusion equation (but it is also known under other names). You may find more information in the above linked wikipedia page, but, in brief, it is an equation which combines a convective cause of time variation at one point (the $\nabla (\vec b \rho)$ term), with a diffusive process, controlled by the Laplacian term.
The convective term in the continuity equation captures the information about the flux of the quantity $\rho$ which locally moves with velocity $\vec b$. When integrated over a closed surface, the net time variation of $\int \rho$ over the volume is due to the surface integral of the current $\rho \vec b $.
The diffusive term with the Laplacian, provides a different mechanism for the net time variation of $\int \rho$ over the volume: the presence of a diffusive flux, not accompanied by a macroscopic field of velocity, proportional to the local gradient of $\rho$, according to a Fick's-like law ( current = $-D \nabla \rho$). 
A: It looks like a diffusion equation with advection. Such an equation would be relevent for heat transport in a fluid moving at velocty ${\bf b}$. If $D=0$ the LHS says that the stuff whose density is  $\rho$ is being moved about the flow. If ${\bf b}=0$ the suff is just diffusing. With both ${\bf b}$ and $D$ non zero, you have a combination of both processes. 
