Global existence for globally Lipschitz vector fields

Builds on Picard–Lindelöf theorem Lipschitz condition Maximal solution Initial value problem

Statement

Let $J$ be an interval, let $K \ge 0$ , and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be continuous. Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in \mathbb{R}^n$ . Then for each $t_0 \in J$ and $x_0 \in \mathbb{R}^n$ , the initial value problem $x' = f(t, x)$ , $x(t_0) = x_0$ has exactly one solution defined on all of $J$ .

Remarks

The point is that the solution does not blow up: the Lipschitz bound gives $\|f(t, x)\| \le \|f(t, 0)\| + K \|x\|$ , so $\|x(t)\|$ grows at most exponentially. A linear system $x' = A(t) x + b(t)$ with continuous $A$ and $b$ on $J$ satisfies the hypothesis on each compact subinterval. So its solutions exist on all of $J$ . A field that is only locally Lipschitz can give blow-up in finite time.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Lipschitz condition definition
Grönwall's inequality statement
Solution of an ordinary differential equation definition
Initial value problem definition
Picard–Lindelöf hypotheses definition
Uniqueness of solutions for Lipschitz vector fields statement
Picard operator definition
Maximal solution definition
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Picard–Lindelöf theorem statement
Global existence for globally Lipschitz vector fields statement

Tag	`0BFD3A6CRH`
URI	`m2://0BFD3A6CRH`
Kind	statement, theorem
Writing	W0 stub
MSC	34A12
Status	This copy of the site has no build data. Run `m2 status 0BFD3A6CRH`.
Uses	Picard–Lindelöf theorem, Lipschitz condition, Maximal solution, Initial value problem

Global existence for globally Lipschitz vector fields

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here