Doing some physics, I've stumbled upon a following equation: $$\operatorname{div}\vec{u} + (\vec{v}, \vec{u}) = 0$$
Where $\vec{v}(x, y, z)$ is known. If it gives any help, $\vec{v}(x, y, z) = \operatorname{grad}g(x, y, z)$ and $\vec{u}(x, y, z) = \operatorname{grad}f(x, y, z)$, where $g(x, y, z) \geq 0$ is known real-valued scalar function and $f(x, y, z)$ is an unknown scalar real-valued function.
I would like to obtain a more or less explicit form of $u$, i.e. express $u$ in terms of $v$.
In the case of 1D, the equation simplifies to $$u' + vu = 0$$ Which is obviously solved by separation of variables and yields a single-parameter set of solutions $$u(x) = C e^{-\int v(x)\,dx}$$
In 3D, solving is far from being straightgorward. It seems that modifying the Bogovskii operator may be not senseless (say Geißert, 2006), but I don't have much experience in the area and I'm afraid of taking wrong direction from the beginning.
I would appreciate for any hints on methods of solving this, as well any references to the books or articles on close topics.