I'm presented with the following internal energy equation: $$ \frac{\partial (\rho e)}{\partial t} + \nabla \cdot (\rho e \vec{v}) = -p\nabla \cdot \vec{v} + \nabla \cdot (K\nabla T) + \varepsilon_V + q_H $$ and I'm being asked to say if it is possible to rewrite it in the form of a conservation law, and if yes, to identify fluxes and sources.
I am making the assumption that "conservation law form" means "in a conservative form". In that case, I would need to write something like: $$ \frac{\partial (\rho e)}{\partial t} + \nabla \cdot \vec{F} = Q $$
Where the $\vec{F}$ term contains the fluxes and Q the sources. The $p\nabla \cdot \vec{v}$ is giving me trouble, as I can't easily rearrange it as the divergence of a vector. I tried writing that $\nabla \cdot p\vec{v} = p\nabla \cdot \vec{v} + \vec{v}\cdot\nabla p $ but that doesn't help.
Is the answer to the question "it can't be done"? And if so, what would you formally say is the reason for this? How would you describe the $p\nabla \cdot \vec{v}$ term?