Global Solutions of Ordinary Differential Equations

Question

Background

Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,

1. $f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
2. for every compact $K \subseteq {\mathbb R}^n$ and every $b > a \geqslant 0$, $$ \int_a^b \|f(t,\cdot)\|_K\,dt < \infty\,, $$ where $$ \|f(t,\cdot)\|_K := \sup_{x \in K} |f(t,x)| + \sup_{\substack{x,y \in K \\ x \neq y}} \frac{|f(x) - f(y)|}{|x - y|}\,, $$ and
3. there exist locally integrable $\alpha, \beta: [0, \infty) \rightarrow [0, \infty)$ such that $$|f(t,x)| \leqslant \alpha(t)|x| + \beta(t)$$ for every $(t,x) \in [0,\infty) \times {\mathbb R}^n$.

We know that, under these hypotheses, the ordinary differential equation $$ x' = f(t,x) $$ generates a unique global flow $$ \varphi: [0,\infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n\,. $$

Question

I'm wondering whether there is a canonical way of realizing $\varphi$ as the limit of globally defined maps $$ \varphi_k: [0,\infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n\,, \quad k \in {\mathbb N}\,. $$

Progress

1. One may assume without loss of generality that $f(t,\cdot)$ is compactly supported uniformly in $t$; that is, there exists a compact $K \subseteq {\mathbb R}^n$ such that $f(t,x) = 0$ whenever $x \notin K$.
2. Setting $\varphi_0(t,x) := x$, then $$ \varphi_k(t,x) := x + \int_0^t f(s, \varphi_{k-1}(s,x))\,ds\,, \quad k = 1, 2, 3, \ldots\,, $$ recursively, as in the standard proof of existence, doesn't seem to work.
3. Euler's proof of existence seems to almost work. But the construction is not "canonical" in the sense that the subsequence along which convergence occurs via Arzela-Ascoli will depend on $f$. I would like to have a sequence $(\varphi_k)_{k \in {\mathbb N}}$ which could be defined by a procedure independent of $f$. This is because I'm ultimately interested in the flow of parametrized differential equations, and I don't want to have to choose a different subsequence for each parameter.

Answer 1

I think I got it! I turned out to be mistaken about item 2 in my Progress Notes.

Lemma 1. Suppose $f \colon {\mathbb R}_{\geqslant 0} \rightarrow {\mathbb R}_{\geqslant 0}$ is locally integrable, and let $F\colon {\mathbb R}_{\geqslant 0} \rightarrow {\mathbb R}_{\geqslant 0}$ be the primitive given by $$ F(t) := \int_0^t f(s)\,ds\,, \quad t \geqslant 0\,. $$ For any positive integer $m$, set $f\colon {\mathbb R}_{\geqslant 0}^m \rightarrow {\mathbb R}_{\geqslant 0}$ by $$ f_m(t) := \prod_{i = 1}^m f(t_j)\,, \quad t = (t_1,\ldots,t_m) \in {\mathbb R}_{\geqslant 0}^m\,, $$ and set $$ S_m(T) := \{t \in {\mathbb R}_{\geqslant 0}^m\,;\ 0 \leqslant t_1 \leqslant \cdots \leqslant t_m \leqslant T\}\,, \quad T \geqslant 0\,. $$ Then $$ \int_{S_m(T)} f_m(t)\,dt = \frac{[F(T)]^m}{m!}\,, \quad \forall T \geqslant 0\,, \quad \forall m = 1, 2, 3, \ldots\,. $$

Proof. Follows by Fubini and integration by substitution. (Details/clarification upon request.)

Assuming that $f(t,\cdot)$ is compactly supported uniformly in $t$ as described in item 1 of my Progress Notes, we may apply Lemma 1 to show that $(\varphi_k(\cdot,x))_{k \geqslant 0}$ is a Cauchy sequence on $[0,T]$ for each $T \geqslant 0$. Thus $$ \varphi(t,x) = \lim_{k \to \infty} \varphi_k(t,x)\,, \quad \forall (t,x) \in {\mathbb R}_{\geqslant 0} \times {\mathbb R}^n\,. $$