Let $\vec j$ be a smooth compactly supported vector field in $\mathbb{R}^3$ such that $div(\vec j)=0$. Are there known either necessary or sufficient conditions such that there exists a sequence of smooth compactly supported vector fields $\{\vec j_i\}$ which converges to $\vec j$ say in $C^2$ topology, $div(\vec j_i)=0$, and such that all the trajectories of each $j_i$ are closed.
If $C^2$ topology can be replaced by something else, that might be of interest too.
