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)


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