= Characteristic representation of a measure-valued continuity equation
{title2=$\rho_t=X(t;0,\cdot)_\#\rho_0$}
Let $b(t,x)$ be continuous in time and continuously differentiable in space, with a uniform spatial <Lipschitz constant> and at most linear growth on each compact time interval. Its <characteristic flow map> exists forward and backward over every bounded time interval. For each initial <probability measure> with finite first <absolute moment>, the displayed <pushforward measure> is the unique <weak solution> of the <continuity equation> $\partial_t\rho+\operatorname{div}(b\rho)=0$ continuous in the first <Wasserstein distance>.
The <Gronwall inequality> bounds the flow by $C_T(1+|x|)$ and therefore preserves finite first <absolute moments>. The <chain rule> establishes the weak equation. For uniqueness, transport a compactly supported terminal <test function> backward along the flow; it solves $\partial_t\varphi+b\cdot\nabla\varphi=0$ and remains compactly supported on a bounded time interval. Inserting it into the weak identity identifies every terminal integral with that of the claimed <pushforward measure>. A zero <divergence> makes the corresponding scalar <transport equation> and the <continuity equation> coincide.
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