Standing regularity of the page's force models checked: uniform field, linear drag and the inverse-square model on its singularity-excluded domain

Model 1: the constant map $z\mapsto m\mathbf{g}$ is continuous by [L3]; and for any two states $\parallel F(t,z)-F(t,z^{\mathrm{\prime }}){\parallel }_2=\parallel m\mathbf{g}-m\mathbf{g}{\parallel }_2=0$, so $L=0$ is a state-Lipschitz constant on every cylinder, independent of time. Hence model 1 satisfies [L1].

Model 2: the map $(t,x,v)\mapsto m\mathbf{g}-cv$ is a constant plus a scalar multiple of the linear map $v\mapsto v$, hence continuous by [L3]; and for states $z=(x,v)$, $z^{\mathrm{\prime }}=(x^{\mathrm{\prime }},v^{\mathrm{\prime }})$, $\parallel F(t,z)-F(t,z^{\mathrm{\prime }}){\parallel }_2=\parallel c(v^{\mathrm{\prime }}-v){\parallel }_2=c\parallel v-v^{\mathrm{\prime }}{\parallel }_2\le c\parallel z-z^{\mathrm{\prime }}{\parallel }_2$ by [L2], so $L=c$ is a state-Lipschitz constant on every cylinder and does not depend on time; in particular model 2 satisfies [L1].

Model 3, Lipschitz bound on an annulus: let $0<r\le R$ and let $x,y\in {\mathbb{R}}^3$ with $r\le \mid x\mid ,\mid y\mid \le R$. Then $\parallel \frac{x}{\mid x{\mid }^3}-\frac{y}{\mid y{\mid }^3}{\parallel }_2\le \frac{\parallel x-y{\parallel }_2}{\mid x{\mid }^3}+\mid y\mid \mid \mid x{\mid }^{-3}-\mid y{\mid }^{-3}\mid \le \frac{\parallel x-y{\parallel }_2}{r^3}+R\cdot \frac{3R^2}{r^6}\parallel x-y{\parallel }_2,$ where the first inequality inserts $y\mathrm{/}\mid x{\mid }^3$ and uses the triangle inequality, and the second uses $\mid \mid x{\mid }^{-3}-\mid y{\mid }^{-3}\mid \le 3R^2{r}^{-6}\mid \mid x\mid -\mid y\mid \mid \le 3R^2{r}^{-6}\parallel x-y{\parallel }_2$, the factorization $\mid v^3-u^3\mid =\mid v-u\mid (v^2+vu+u^2)$ applied to $u=\mid x\mid$, $v=\mid y\mid$ and the reverse triangle inequality of [L2]; multiplying by $k$ gives the constant $L=k(r^{-3}+3R^3{r}^{-6})$ for $F=-kx\mathrm{/}\mid x{\mid }^3$ on the annulus (the state variable enters only through $x$).

Model 3, continuity and local-Lipschitz on the domain: every nonempty compact subset $K$ of $Q={\mathbb{R}}^3\setminus \{0\}$ lies in an annulus $r\le \mid x\mid \le R$ with $r={\min }_K\mid x\mid >0$ and $R={\max }_K\mathrm{\mid }x\mathrm{\mid }<\mathrm{\infty }$ by [L2] and [L4] (the norm is continuous by its reverse triangle inequality, and its attained minimum is positive because $0\mathrm{\notin }K$); by step 1.3 the map is Lipschitz with the finite constant above on each such annulus, the empty compact set imposes no inequality, and small closed balls around each point of $Q$ avoid the external source, hence the map is locally Lipschitz, and in particular continuous, on $Q$; the time variable does not occur, so the constant is locally uniform in time. Model 3 therefore satisfies [L1] on its domain.

Model 3, no extension to the excluded point: for $x\mathrm{\ne }0$ one has $\parallel F(x){\parallel }_2=k\mid x{\mid }^{-2}$, so for every bound $M>0$ there is $\delta =\sqrt{k\mathrm{/}M}>0$ with $\parallel F(x){\parallel }_2>M$ whenever $0<\mid x\mid <\delta$; hence no continuous extension of $F$ to $x=0$ exists (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point: a continuous extension would carry a sequence $x_n\to 0$ to a convergent image sequence, but the values are unbounded). The external-source singularity exclusion $Q={\mathbb{R}}^3\setminus \{0\}$ is therefore not cosmetic: it is what places model 3 inside the force branch at all.

Conclusion: models 1-3 satisfy the standing hypotheses of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, with the constants computed above; hence the associated first-order fields of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model satisfy the Picard–Lindelöf hypotheses and the local existence, uniqueness and maximal-motion conclusions of Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder, as well as the continuation criterion of Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain, apply to each of them on its stated domain.