Lemma: Spatial rod realization

Equations (eq:Y1)–(eq:Y2) give unique smooth decaying rod coefficients on the stated classical domain. They are $O(\text{abs}{\mathbf{x}}^{-2})$ on compact time intervals.

Proof. Both sources are divergences of compact smooth spatial vectors. For the second source, $B_1\partial Y_1$ is compact because $B_1$ is compact. Newton convolution gives smooth solutions. Integration by parts transfers one derivative to the Newton kernel and yields the $1/r^2$ bound.

The difference of two decaying solutions is entire harmonic and tends to zero at infinity. The maximum principle on successively larger balls makes it zero.