Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous [continuity equation](https://en.wikipedia.org/wiki/Continuity_equation) $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact support (in both space and time)? What are the simplest nontrivial solutions with compact support? 



On solutions of the continuity equation
If there is no connection between $\rho$ and $j$, you may as well combine them into a 4d vector $F=(\rho,j)$. If I understand it correctly, the question is: If $F$ is square integrable and divergence-free, can it be approximated by divergence-free vector fields of compact support? This is true and can be shown along the following lines: First define an approximation $F\_\epsilon$ by truncating the Fourier transform of $F$ in a neighborhood of the origin. $F\_\epsilon$ is still divergence-free. Next, use the theory of differential forms to represent $F\_\epsilon$ as $\nabla\wedge A\_\epsilon$, with a tensor-valued potential $A\_\epsilon$. Since $F\_\epsilon$ vanishes in a neighborhood of the origin, you can pick $A\_\epsilon$ to be square integrable. Finally, simply truncate $A\_\epsilon$ to get compact support. 


To get nontrivial examples, just take $F=\nabla\wedge A$, where $A$ has compact support.


