Theorem: Uniform Newton-divergence tail

Suppose $b^i(t,\mathbf{x})$ is smooth and either:

1. has uniformly compact spatial support for $t\in J$, with all required derivatives bounded; or

2. has finite seminorms (eq:Schwartz) for every $N,m,\alpha$.

Then

is a smooth decaying solution of $\Delta u=-{\partial }_i{b}^i$, and

outside a fixed ball. It is the unique smooth solution with $u\to 0$ at spatial infinity.

Proof. Integration by parts gives

For compact support, the kernel and its spatial derivatives are bounded by constants times $|\mathbf{x}{|}^{-2-|\alpha |}$ once $|\mathbf{x}|$ is twice the support radius. Time differentiation acts only on $b$.

For Schwartz sources, split the integral into $|\mathbf{y}|\le |\mathbf{x}|/2$ and its complement. The first part has the same kernel bound, using a uniform $L^1$ bound on the source. In the complementary part, use a smooth partition around $\mathbf{y}=\mathbf{x}$; transfer derivatives of the singular kernel onto the smooth rapidly decreasing source when necessary. The source and its derivatives decrease to arbitrary order in that region, while the dipole kernel is locally integrable. This bounds that part to an arbitrarily high inverse power after increasing the Schwartz seminorms used in the estimate.

The same argument proves smoothness and permits time differentiation. A difference of two decaying solutions is an entire harmonic function tending to zero. The maximum principle gives uniqueness.