Besov spaces and self-similar solutions for the wave-map equation

• Finally (and this is somehow the dual of the previous point), infinite energy data
enable us to consider (covariant) wave-maps such that φ(0) is not necessarily the
north or the south pole of the sphere S d , so that a singularity occurs at 0.
Plan of the article
In Section 2, we will prove existence, uniqueness and scattering of solutions for data that
are of infinite energy but small. We shall use a classical fixed-point argument.
In Section 3, we will study the case of large data. However, since we cannot resort to
perturbation techniques anymore, we will have to restrict to the case of self-similar data.
Matters reduce then to studying a (singular) ODE.
In Section 4, we shall see that we can apply the results of the last two sections in order
to describe a little more precisely the blow-up phenomenon for the wave-map equation on
the sphere S 3 .
1.4 Littlewood-Paley decomposition and associated spaces
We shall present in this section the main harmonic analysis tools that will be used in the
sequel.
A Littlewood-Paley decomposition is defined in the following way: consider Ψ a function
supported in the annulus centered in 0, of small radius 3/4, and big radius 8/3, such that
ξ
Ψ
2j
= 1 for ξ = 0 .
j∈Z
Then the Fourier multipliers ∆j and Sj are defined by
D
∆j = Ψ
2j
SN = 1 −
D
Ψ
2j
j≥N+1
Thus any distribution can be decomposed (modulo polynomials) into a sum of elementary
elements that are localized in frequency:
f =
∆jf or f = SNf +
∆jf .
j∈Z
We can now define the (homogeneous) Besov spaces ˙ B s q,r
f ˙ B s q,r
def
=
2 js ∆jf q
L
x ℓr j
.
j≥N+1
which are given by the norm
Notice that the homogeneous L 2 -based Sobolev spaces identify with Besov spaces
˙H s = ˙ B s 2,2 .
A variant of Besov spaces which are particularly well suited for the study of PDEs is given
by the spaces L p (R, ˙ B s q,r ) - we will later omit the R in order to make the notations lighter.
4
.