 Solution

The characteristic equations for a transport equation on phase space areA sufficient global hypothesis is that $F$ is continuous in time, locally Lipschitz continuous in $(x,v)$ uniformly on compact time intervals, and, for every finite $T$, satisfies $|F(t,x,v)|\le C_T(1+|x|+|v|)$ for $|t|\le T$. These hypotheses give unique characteristic curves for all real times if the field is defined for all real times. It suffices for forward existence to impose them on nonnegative time. One may strengthen the spatial hypothesis to $C^1$ when differentiating the characteristic flow map below. Local Lipschitz continuity alone guarantees only local existence; the linear-growth bound prevents finite-time escape.