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A bounded vector field on all of Euclidean space is complete

Statement

Let $V : R^{n} \to R^{n}$ be a bounded continuous vector field that is locally Lipschitz in the state variable. Then every maximal solution of the autonomous ODE $x^{'} = V (x)$ is defined for all time. In particular, a bounded smooth Euclidean vector field is complete.

Facts & Assumptions

Given: A bound $∥ V (x) ∥_{2} \leq M$ for all $x \in R^{n}$ and a maximal solution $x : (α, β) \to R^{n}$ of $x^{'} = V (x)$ .

[F1]

This is an autonomous ODE in the sense of Autonomous ordinary differential equations.

[F2]

A solution satisfies the Volterra integral equation (A first-order initial value problem is equivalent to its Volterra integral equation).

[L1]

The norm of a vector-valued integral is at most the integral of the norm (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[L2]

If a maximal solution had a finite endpoint, then near that endpoint it would leave every compact subset of the ODE domain (At a finite maximal time an ODE solution leaves every compact subset of the domain).

Proof

technique · direct

1.1

Fix $t_{*} \in (α, β)$ . If $t \geq t_{*}$ , then [F2] and [L1] give the first inequality below, and if $t \leq t_{*}$ they give the reflected inequality.

F2L1

$∥ x (t) - x (t_{*}) ∥_{2} = {∥ \int_{t_{*}}^{t} V (x (s)) d s ∥}_{2} \leq \int_{t_{*}}^{t} ∥ V (x (s)) ∥_{2} d s \leq M (t - t_{*}) .$

If $t \leq t_{*}$ , the oriented Volterra equation gives $x (t) - x (t_{*}) = - \int_{t}^{t_{*}} V (x (s)) d s$ , so [L1] yields

$∥ x (t) - x (t_{*}) ∥_{2} \leq \int_{t}^{t_{*}} ∥ V (x (s)) ∥_{2} d s \leq M (t_{*} - t) .$

Thus $∥ x (t) - x (t_{*}) ∥_{2} \leq M ∣ t - t_{*} ∣$ for every $t \in (α, β)$ , so on every finite time interval the solution stays in one Euclidean ball about $x (t_{*})$ .

2.1

Suppose $β < \infty$ . Then step 1.1 shows that for $t$ close to $β$ the graph point $(t, x (t))$ stays inside the compact box $[β - 1, β] \times \overline{B} (x (t_{*}), M (β - t_{*}) + 1)$ of the ODE domain $R \times R^{n}$ , contradicting [L2]. Thus $β = \infty$ . The same argument at the left endpoint gives $α = - \infty$ .

L2step 1.1

3.1

Therefore every maximal solution is global, so the vector field is complete.

F1step 2.1∎

Depends on

Used by

A compactly supported smooth Euclidean vector field is complete Corollary

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nigel Hitchin, Differentiable Manifolds, Appendix §10.2 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)