Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous 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?
2 Answers
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.
If $j$ has compact support in $\mathbb R_t\times\mathbb R^3_x$, $\rho$ will have compact support iff $\int_{-\infty}^\infty\nabla\cdot j_s\ ds\equiv0$.
The set of approximable $L^2$ solutions should be the closure of these.
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$\begingroup$ $\rho_t(x)=-\int_{-\infty}^{t} \nabla\cdot j_s\ ds$, so the support of $\rho$ is contained in the convex hull of the support of $\nabla\cdot j$, which is itself contained in the convex hull of the support of $j$. $\endgroup$ Apr 24, 2018 at 15:44
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$\begingroup$ But this only gives compact support of $\rho$ at each fixed time. I was asking for compactness in 4 dimensions. This makes the question nontrivial. $\endgroup$ Apr 24, 2018 at 15:46
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$\begingroup$ If $j$ vanishes outside $[-A,A]\times B_R$ (and the integral condition holds), so does $\rho$. Isn't it obvious ?? $\endgroup$ Apr 25, 2018 at 15:48