In rotational dynamics the Newton-Euler equations express the dependence of angular velocity $\boldsymbol{\omega}$ of a rigid body from the torque $\boldsymbol{\tau}_{cm}$ with respect to the centre of mass $\boldsymbol{\tau}_{cm}=I_{cm}\frac{d\boldsymbol{\omega}}{dt}+\boldsymbol{\omega}\times(I_{cm}\boldsymbol{\omega})$, where $I_{cm}$ is the inertia matrix with respect to the centre of mass, which is invertible, so that$$\frac{d\boldsymbol{\omega}}{dt}=-I_{cm}^{-1}(\boldsymbol{\omega}\times(I_{cm}\boldsymbol{\omega}))+I_{cm}^{-1}\boldsymbol{\tau}_{cm}.$$
My question is: given an initial value $\boldsymbol{\omega}(t_0)=\boldsymbol{\omega}_0$, can the global existence and uniqueness of the solution $\boldsymbol{\omega}$ to such a problem be guaranteed?
From the following theorem of existence and uniqueness
Let $A\subset\mathbb{R}\times\mathbb{R}^n$ be an open set and let $\boldsymbol{f}=\boldsymbol{f}(t,\boldsymbol{y}):A\subset\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ be a continuous function, $\boldsymbol{f}\in C(A)$, with continuous partial derivatives $\frac{\partial\boldsymbol{f}}{\partial y_i}\in C(A)$, $i=1,\ldots,n$.
If $(t_0,\boldsymbol{y}_0)\in A$ there there is a $\delta>0$ such that the initial value problem$$\begin{cases} \boldsymbol{y}'=\boldsymbol{f}(\boldsymbol{y},t) \\\boldsymbol{y}(t_0)=\boldsymbol{y}_0\end{cases}$$has one and only one solution defined on the interval $[t_0-\delta,t_0+\delta]$.
by taking into account that $\boldsymbol{f}(\boldsymbol{\omega},t)=-I_{cm}^{-1}(t)(\boldsymbol{\omega}\times(I_{cm}(t)\boldsymbol{\omega}))+I_{cm}^{-1}(t)\boldsymbol{\tau}_{cm}(t)$ is such that, if my calculations are not wrong, $\frac{\partial\boldsymbol{f}}{\partial \omega_i}\in C(\mathbb{R}^{3}\times \mathbb{R})$, I see that for any $t_0\in\mathbb{R}$ there always exist a locally unique solution to $\frac{d\boldsymbol{\omega}}{dt}=-I_{cm}^{-1}(\boldsymbol{\omega}\times(I_{cm}\boldsymbol{\omega}))+I_{cm}^{-1}\boldsymbol{\tau}_{cm}$, $\boldsymbol{\omega}(t_0)=\boldsymbol{\omega}_0$, but can we guarantee that one and only one solution $\boldsymbol{\omega}$ exists defined on all $\mathbb{ R}$ ? I heartily thank any answerer!