My title - Departamento de Matemática da Universidade do Minho

6 FLOWS 47
Uniqueness of solutons. A velocity field v(t, x), defined in a domain I × D of the extended
phase space R × R n , is locally Lipschitz w.r.t. to the variable x if for any (t 0 , x 0 ) ∈ I × D there is
a neighborhood J × U ∋ (t 0 , x 0 ) and a constant L ≥ 0 such that
‖v(t, x) − v(t, y)‖ ≤ L · ‖x − y‖
∀ (t, x), (t, y) ∈ J × U
If v(t, x) has continuous derivative w.r.t. x, i.e. if the Jacobian
( ) ∂vi
D x v(t, x) = (t, x)
∂x j
exists and is continuous, then v(t, x) is locally Lipschitz in any compact convex domain I × K ⊂
R × R n . The basic uniqueness theorem is 14
Picard-Lindelöf theorem. Let v(t, x) be a continuous velocity field defined in some domain D
of the extended phase space R×X. If v is locally Lipschitz (for example continuously differentiable)
w.r.t. the second variable x, then there exist one and only one local solution of ẋ = v(t, x) passing
through any point (t 0 , x 0 ) ∈ D.
Geometrically, the uniqueness theorem says that through any point (t 0 , x 0 ) of the domain D
there pass one and only one solution. Hence solutions, considered as curves in the extended phase
space, cannot intersect each other.
In a domain where Picard’s theorem applies, if two local solutions agree in a common interval
of times then they are indeed restrictions of a unique solution defined in the union of the respective
domains. There follows that solutions are always extendible to a maximum domain. Such solutions
are called maximal solutions.
Strategy of the proof of the Picard’s theorem. The first observation is that a function
ϕ(t) is a solution of the Cauchy problem for ẋ = v(t, x) with initial condition ϕ(t 0 ) = x 0 iff
∫ t
ϕ(t) = x 0 + v (s, ϕ(s)) ds
t 0
Now, we notice that the above identity is equivalent to the statement that ϕ is a fixed point of the
so called Picard’s map φ ↦→ Pφ, sending a function t ↦→ φ(t) into the function
∫ t
(Pφ) (t) = x 0 + v (s, φ(s)) ds
t 0
At this point, one must chose cleverly the domain of the Picard’s map, which is the space of
functions where we think a solution should be. It will be a certain space C of continuous functions,
defined in an appropriate neighborhood I of t 0 , equipped with a norm that makes it a complete
metric space (hence a Banach space). The Lipschitz condition, together with continuity, satisfied
by the velocity field will imply that if the interval I is sufficiently small then the Picard’s map
P : C → C is a contraction. The contraction principle (a.k.a. Banach fixed point theorem) finally
guarantees the existence and uniqueness of the fixed point of P in C.
Picard’s iterations. The contraction principle actually says that the fixed point, i.e. the
solution we are looking for, is the limit of any sequence φ, Pφ, ..., P n φ, ... of iterates of the Picard
map starting with any initial guess φ ∈ C. In other words, the existence part of the theorem is
“constructive”, it gives us a procedure to find out the solution, or at least a sequence of functions
which approximate the solution.
14 M. 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 114 (1894),
454-457. Digitized version online via http://gallica.bnf.fr/ark:/12148/bpt6k3074