# Solution of equation on vector field

I have a vector field function $$\vec{J}: {\bf R}^3\to {\bf R}^3$$ looking like:

$$\vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r})$$

with a (very well behaved) real, positive, differentiable scalar function $$\rho: {\bf R}^3\to {\bf R}^+$$ and the equation should hold for all non-zero constant $${\bf R}^3$$ vectors $$\vec{B}\ne\vec{0}$$. For physical reasons I am out for the set of vector field functions $$\vec{v}(\vec{r})$$ such that

$$\nabla \cdot \vec{J}(\vec{r}) = 0,$$

i.e. $$\vec{J}$$ gets divergence free in all points $$\vec{r}$$ in $$\bf R^3$$ (one specific $$\vec{v}(\vec{r})$$ should fulfill the condition for all $$\vec{B}$$). By simplification of the equation (looking at special joices of $$\vec{B}$$) I could guess a set of solutions

$$\vec{v}_F(\vec{r})=\nabla F(\rho(\vec{r}))=f(\vec{r}) \nabla(\rho(\vec{r}))$$

i.e. effectively all vector fields that are parallel to $$\nabla \rho$$.

I conjecture that there are no other fields than $$\vec{v}_F(\vec{r})$$ such that the continuity is fulfilled. I like to proof that but fail. A "systematic" solution involves a singular linear equation system which seems to be a bit above my humble abilities to really handle with full oversight properly in a systematic manner (praying to god that this won't disqualify me to ask here), but I thought of trying something like a proof by contradiction by inserting a (scaled) component $$\vec{v}_c{(\vec{r})}$$ that is orthogonal to $$\nabla \rho$$ and assumed to be non-zero at least somewhere $$(\vec{r}_0)$$ and for all $$\vec{B}$$:

$$\vec{v}_c(\vec{r}_0) = \nabla \rho(\vec{r}_0) \times (\nabla \rho(\vec{r}_0) \times \vec{v}(\vec{r}_0))$$

But I fail to bring that to an happy end, since I end with three terms that might be non-zero but I do not know how to show that they eventually can't get zero when added up. A solution would be greatly appreciated.

(I feel that this is a border line case between MSE and MO, however since I am mostly out for the solution and since its a little but important puzzle stone in bigger research project, I finally decided to post it here)