View source for Cauchy-Lipschitz theorem

{{MSC|34A12}} 
[[Category:Ordinary differential equations]]
{{TEX|done}}

One of the existence theorems for solutions of an ordinary  differential equation (cf. [[Differential equation,  ordinary|Differential equation, ordinary]]), also called Picard-Lindelof theorem or Picard existence theorem by some authors. The theorem concerns the initial value problem
\begin{equation}\label{e:IVP}
\left\{
\begin{array}{ll}
\dot{x} (t) = f(x(t), t)\\
x (0) = x_0\, 
\end{array}\right.
\end{equation}
(a solution of \eqref{e:IVP} is often called an [[Integral curve|integral curve]] of $f$ through $x_0$).
A version (the local one) of the theorem states the following

'''Theorem 1'''
Let $U\subset \mathbb R^n$ be an open set and $f: U\times [0,T] \to \mathbb R^n$ a continuous function which satisfies the [[Lipschitz condition]]
\begin{equation}\label{e:Lipschitz}
|f (x_1, t) - f (x_2, t)|\leq M |x_1-x_2| \qquad \forall (x_1, t), (x_2, t)\in U \times [0,T]\, 
\end{equation}
(where $M$ is a given constant). If $x_0\in U$, then for some positive $\delta$ there is a ''unique solution'' $x: [0,\delta]\to U$ of the initial value problem \eqref{e:IVP}.

A similar statement holds if $[0,T]$ is replaced by $[-T, 0]$ or $[-T, T]$ (the interval of existence becomes then, respectively, 
$[-\delta, 0]$ and $[-\delta, \delta]$). The existence is limited to a small interval because the integral curve $x$ might "leave" the domain $U$. In particular, either $\delta$ can be taken equal to $T$ or there is a maximal time interval of existence $[0, T_0[$ of the solution, characterized by the property that $x(t)$ approaches the boundary of $U$ as $t\to T_0$. 

A global version of the same statement is the following

'''Theorem 2'''
Let $U = \mathbb R^n$ and $f$ as in Theorem 1. Then, for any $x_0\in \mathbb R^n$ there is a unique solution $x:[0, T]\to U$ of the initial value problem \eqref{e:IVP}.

The global Theorem 2 holds also when $[0,T]$ is replaced by $[-T, 0]$ or $[-T,T]$, by the unbounded halflines $[0, \infty[$ and $]-\infty, 0]$) or by the entire real line $\mathbb R$. The existence part of the statement holds under much weaker conditions: see for instance [[Peano theorem]] and [[Caratheodory conditions]]. The uniqueness can only be improved slightly, see [[Osgood criterion]]. 

Both theorems 1 and 2 are used to derive the existence (and uniqueness) of integral curves of vector fields on manifolds, under appropriate regularity assumptions. In fact the local existence of an integral curve $\dot{\gamma} (t) = X (\gamma (t))$ where $X$ is a vector field on the [[Tangent bundle|tangent bundle]] of a [[Differentiable manifold]] reduces to the initial value problem \eqref{e:IVP} in local charts. 

====Flows of vector fields and one-parameter families of diffeomorphisms====
A strenghtened version of the Cauchy-Lipschiz theorem is of great importance in several branches of mathematics (differential topology, dinamical systems and Lie theory among others).

'''Theorem 3'''
Let $f$ be as in Theorem 2 and for each $t\in [0,T]$ consider the map $\Phi_t : \mathbb R^n \to \mathbb R^n$ given by $\Phi_t (x_0) = x (t)$, where $x$ is the unique solution of \eqref{e:IVP}. Then $\Phi_t$ is a biLipschitz homeomorphisms of $\mathbb R^n$ onto itself, more precisely it satisfies 
\[
e^{-Mt} |x_1-x_2| \leq |\Phi_t (x_1)-\Phi_t (x_2)| \leq e^{Mt} |x_1-x_2| \qquad \forall x_1, x_2\in \mathbb R^n\, .
\]

Theorem 3 is a simple consequence of Theorem 2 and [[Gronwall lemma]].
As for the other theorems, similar versions hold for different time intervals.

The theorem above holds also:
* for $C^1$ vector fields when $\mathbb R^n$ is substituted by a compact differentiable manifold;
* for Lipschitz vector fields when $\mathbb R^n$ is substituted by an open set $U\subset \mathbb R^n$ and it is known apriori that integral curves of $f$ through any $x_0\in U$ do not "exit" the domain $U$ before the time $t$ (namely the maximal interval of existence contains $t$ for any initial value $x_0$). 
Similarly, one can state an appropriate version on open subsets of manifolds.

If $f$ is, in addition, $C^1$ (resp. $C^k$, $C^\infty$ or $C^\omega$), the map $\Phi_t$ is a $C^1$ (resp. $C^k$, $C^\infty$ or $C^\omega$) diffeomorphism. The family $\{\Phi_t\}$ is called a one-parameter family of diffeomorphisms and the map $\Phi (t,x_0):= \Phi_t (x_0)$ is the flow of the (time-dependent) vector field $f$. When $f$ is independent of $t$, the family (which is then defined for every $t\in \mathbb R$) is said to be ''generated'' by the vector field $f$.

====References====
{|
|-
|valign="top"|{{Ref|Am}}||  H. Amann, "Ordinary differential equations. An introduction to  nonlinear analysis." de Gruyter Studies in Mathematics, 13. Walter de  Gruyter & Co., Berlin,  1990.
|-
|valign="top"|{{Ref|Ha}}|| P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)
|-
|valign="top"|{{Ref|Li}}||  E. Lindelöf, "Sur l'application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre", ''Comptes rendus hebdomadaires des séances de l'Académie des sciences'' '''116''' (1894) pp. 454–457.
|-
|valign="top"|{{Ref|Pet}}|| I.G. Petrovskii,    "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated  from Russian)
|-
|}