As in the title: given a vector field $\vec f$, are there any interesting applications (in physics, biology, or economy, or ...) of the partial differential equation
$ - \operatorname{grad} ( \operatorname{div} \vec u ) = \vec f$
with unknown vector field $\vec u$?
I am aware that the PDE is generally under-constrained in that form. I am just interested in whether this PDE has got any interesting merit on its own.
This is the continuity equation in disguise. Since $\text{curl}\,\vec{f}=0$, we can write $\vec{f}=\text{grad}\,\Phi$ for a scalar function $\Phi$, and then $$-\text{div}\,\vec{u}=\Phi+\text{constant}.$$ If we fix the constant at zero (for example, by considering the $r\rightarrow\infty$ limit) and identify $\Phi=\partial\rho/\partial t$, then this is the continuity equation for a velocity field $\vec{u}(\vec{r},t)$ with density $\rho(\vec{r},t)$.
