= Global characteristic flow under linear growth
If $F\in C^1(\mathbb R\times\mathbb R^d)$ and $|F(t,x)|\le K(1+|x|)$, every <characteristic curve> is defined for all real time. The <Gronwall inequality> bounds $1+|X(t)|$ on each bounded time interval, so local <ordinary differential equation> existence cannot terminate through escape to infinity. Uniqueness and the variational equation give a global $C^1$ spatial flow diffeomorphism, with Jacobian $J>0$ satisfying $J_t=(\operatorname{div}F)(t,X)J$.
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