Orthogonal shell force local IVP and compact continuation

If $K_{\mathcal{S}}:=K\cap \mathcal{S}$ is empty, no shell-force trajectory takes a value in $K$; both implications in the Statement are then immediate. Otherwise $K_{\mathcal{S}}$ is compact, since $\mathcal{S}$ is closed relative to $\mathcal{D}$: the metric norm equation passes to limits, and a unit timelike limit has nonpositive pairing with the time-orientation field as a limit of future vectors; that pairing cannot vanish for a timelike vector, so the limit remains future-directed. For each $w\in K_{\mathcal{S}}$, [L1] supplies a relative neighborhood $V_w$ in $\mathcal{S}$ and ${\delta }_w>0$ such that solutions through all normalized shell states in $V_w$ exist on a common interval $[-{\delta }_w,{\delta }_w]$ about the initial parameter. These neighborhoods cover $K_{\mathcal{S}}$. Compactness gives a finite subcover $V_{w_1},\dots ,V_{w_N}$; put $\delta :={\min }_{1\le j\le N}{\delta }_{w_j}>0$. Since the coordinate system is autonomous, every initial state in $K_{\mathcal{S}}$ has a solution on $[s_0-\delta ,s_0+\delta ]$ about any initial parameter $s_0$. Also, continuity of $G$ gives for each $w\in K$ a neighborhood $O_w$ on which $\mathrm{\parallel }G(\zeta ){\mathrm{\parallel }}_2<\mathrm{\parallel }G(w){\mathrm{\parallel }}_2+1$. A finite subcover of the compact set $K$ therefore gives a finite bound $M$ for $\mathrm{\parallel }G{\mathrm{\parallel }}_2$ on $K$.

If $z([b_0,b))\subseteq K$, then for $b_0\le s<t<b$, [L4], [L2] and [L3] give $\mathrm{\parallel }z(t)-z(s){\mathrm{\parallel }}_2={\parallel {\int }_s^{t}G(z(r))dr\parallel }_2\le {\int }_s^t\mathrm{\parallel }G(z(r)){\mathrm{\parallel }}_{2}dr\le M(t-s)$. For the explicit sequence $t_n:=b-(b-b_0)\mathrm{/}(n+1)\uparrow b$, the sequence $z(t_n)$ is Cauchy. The closures in $K$ of its nonempty tails are nested nonempty closed sets, so compactness gives a point $z_b$ in their intersection. The Cauchy property then gives $z(t_n)\to z_b\in K$: for each $ϵ>0$, choose $N_0$ so the tail has diameter less than $ϵ\mathrm{/}2$, then choose $n\ge N_0$ with $\mathrm{\parallel }z(t_n)-z_b{\mathrm{\parallel }}_2<ϵ\mathrm{/}2$ from the closure property, after which all $k\ge N_0$ satisfy $\mathrm{\parallel }z(t_k)-z_b{\mathrm{\parallel }}_2<ϵ$. The estimate above, with $t_n>t$ and $n\to \mathrm{\infty }$, shows $\mathrm{\parallel }z(t)-z_b{\mathrm{\parallel }}_2\le M(b-t)$, so the full trajectory tends to $z_b$ as $t\uparrow b$. Continuity of the metric and the time-orientation field gives $z_b\in \mathcal{S}$.

Fix a compact $K\subseteq \mathcal{D}$. If $K_{\mathcal{S}}:=K\cap \mathcal{S}$ is empty the solution never enters $K$. Otherwise let $\delta >0$ be the common local existence radius from step 1.1 for $K_{\mathcal{S}}$. If a finite right-endpoint solution did not eventually leave $K$, there would be a time $t_0\in (\max \{a,b-\delta \mathrm{/}2\},b)$ with $z(t_0)\in K$. The local solution through $z(t_0)$ reaches beyond $b$ and agrees with $z$ by uniqueness, contradicting maximality. At a finite left endpoint, non-exit would similarly give $t_0\in (a,\min \{b,a+\delta \mathrm{/}2\})$ with $z(t_0)\in K$, and the local solution would extend before $a$. Thus the solution eventually leaves every compact subset at either finite endpoint.

If $a>-\mathrm{\infty }$ and $z((a,b_0])\subseteq K$, use the explicit sequence $s_n:=a+(b_0-a)\mathrm{/}(n+1)\downarrow a$. For $a<s<t\le b_0$, [L4], [L2] and [L3] give $\mathrm{\parallel }z(t)-z(s){\mathrm{\parallel }}_2={\parallel {\int }_s^{t}G(z(r))dr\parallel }_2\le {\int }_s^t\mathrm{\parallel }G(z(r)){\mathrm{\parallel }}_{2}dr\le M(t-s)$, so $z(s_n)$ is Cauchy. The closures in $K$ of its nonempty tails are nested nonempty closed sets, so compactness gives a point $z_a$ in their intersection; the Cauchy property then gives $z(s_n)\to z_a\in K$. Continuity of the metric and time orientation gives $z_a\in \mathcal{S}$. For each fixed $t\in (a,b_0]$, eventually $s_n<t$, and the same estimate gives $\mathrm{\parallel }z(t)-z_a{\mathrm{\parallel }}_2\le M(t-a)$; hence $z(t)\to z_a$ as $t\downarrow a$. A local solution through $z_a$ at parameter $a$ has derivative $G(z_a)$, matching the right derivative of the continuous extension of $z$; gluing it just before $a$ gives a solution extending the maximal one, by uniqueness in [L1].

By [L1], the same smooth shell-force system has a local solution $\tilde{z}$ through $z_b$ at parameter $b$. The extension of $z$ by $z(b):=z_b$ is continuous. Since $G(z(t))\to G(z_b)$ as $t\uparrow b$, the integral identity in step 2.1 shows that the left derivative at $b$ is $G(z_b)$, equal to ${\tilde{z}}^{\mathrm{\prime }}(b)$. The extended curve therefore glues to $\tilde{z}$ just after $b$ as a solution; uniqueness in [L1] makes it agree with the original on the overlap, contradicting maximality. Hence the right-endpoint solution extends past $b$.

Steps 2.1 and 3.1 prove right-endpoint continuation, step 2.3 proves the left-endpoint analogue, and step 2.2 establishes eventual exit from every compact set.