Compactly supported time-dependent vector fields have global evolution on a compact time interval

Let $\gamma :(a,b)\to M$ be a maximal solution of $\dot{\gamma }(t)=X_t(\gamma (t))$ with initial time $s\in J$. If $\gamma (t_0)\mathrm{\notin }K$ for some $t_0\in (a,b)$, then $X_t(\gamma (t_0))=0$ for every $t\in J$ by [L3], so the constant curve through $\gamma (t_0)$ also solves the equation. Uniqueness therefore forces $\gamma$ to be constant on the connected component of $\{t\in (a,b):\gamma (t)\mathrm{\notin }K\}$ containing $t_0$. Thus every nonconstant part of the trajectory stays inside $K$.

Suppose $b<\sup J$. Choose times $t_n\uparrow b$. If infinitely many $\gamma (t_n)$ lie in $K$, compactness of $K$ gives a subsequence converging to some $q\in K$. Otherwise $\gamma (t_n)\mathrm{\notin }K$ for all large $n$, and step 1.1 makes those tail values constant on a neighbourhood of $b$; hence $\gamma (t_n)\to q$ for some $q\in M\setminus K$. Applying [L1] at $(b,q)$ gives $\epsilon >0$, an open neighbourhood $U$ of $q$, and a local evolution operator ${\mathrm{\Psi }}_{t,s}$ for $s,t\in (b-\epsilon ,b+\epsilon )$. Choose $n$ large enough that $t_n\in (b-\epsilon ,b)$ and $\gamma (t_n)\in U$. Then $t\mapsto {\mathrm{\Psi }}_{t,t_n}(\gamma (t_n))$ is a solution on $(b-\epsilon ,b+\epsilon )$. On the common interval $(b-\epsilon ,b)$ it agrees with $\gamma$ by uniqueness, because both solve the same equation and have the same value at time $t_n$. This extends $\gamma$ past $b$, contradicting maximality.

The same argument excludes a left endpoint larger than $\inf J$. Therefore every maximal solution with initial time in $J$ exists on all of $J$.

Define ${\mathrm{\Psi }}_{t,s}(p)$ to be the value at time $t$ of the unique solution starting from $p$ at time $s$. Step 3.1 makes this global on $J$, and [L2] supplies the cocycle law. Hence ${\mathrm{\Psi }}_{t,s}:M\to M$ is the desired global evolution operator.