Cauchy-Lipschitz theorem

2020 Mathematics Subject Classification: Primary: 34A12 [MSN][ZBL]

One of the existence theorems for solutions of an ordinary differential equation (cf. 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 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 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

[Am]	H. Amann, "Ordinary differential equations. An introduction to nonlinear analysis." de Gruyter Studies in Mathematics, 13. Walter de Gruyter & Co., Berlin, 1990.
[Ha]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[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.
[Pet]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)