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

These spaces were introduced by Chemin and Lerner [5] in the context of PDEs (notice
that very similar spaces were considered earlier by Triebel [14] from a harmonic analysis
point of view). The norm of these spaces is defined as follows
fe Lp (R, B˙ s )
q,r
def
= 2 js ∆jf p
L
t Lq
x ℓr j
In Section 2, we will build up a solution to (3) in Chemin-Lerner spaces.
Let us record the two following results, which extend classical theorems on Besov spaces.
Theorem 1.1 (Sobolev embedding) Suppose that
we have then
q ≥ Q and s − n
q
L p ˙ B S Q,r ↩→ L p ˙ B s q,r .
The proof simply relies on Bernstein inequality.
= S − n
Q ,
Proposition 1.1 Suppose s > 0 and
f =
fk with Supp fk ⊂ B(0, C2 k ) .
One has then
k
f L p (R, ˙ B s q,r) ∼
2 Small data in Besov spaces
2 js fjf L p
t Lq
x ℓr j
2.1 Main result: existence, uniqueness, and scattering
Well-posedness for the wave-map system is only known for data in Sobolev spaces. We
want to extend this to Besov spaces, in the same spirit as Planchon [12] did for the wave
equation with a power non-linearity.
Theorem 2.1 Let n ≥ 5 (i.e. d ≥ 3), and consider data (v0, v1) ∈ ˙ B n/2−1
2,∞ × ˙ B n/2−2
2,∞ radial
and small enough. Then there exists a global solution v of (3), which is unique in a ball
of sufficiently small radius of
(5) X def
∞
= L R, ˙B n
2 −1
∩ L 2
∩ L 3
.
2,∞
R, ˙B n
2 −2
( 1
2
− 3
2n) −1 ,∞
.
.
R, ˙B n
2 −2
( 1
2
− 4
3n) −1 ,∞
Furthermore, a weak scattering takes place, in the sense that there exists (v + 0 , v+ 1 ) and
(v − 0 , v− 1 ) in ˙ B n/2−1
2,∞ × ˙ B n/2−2
2,∞ such that
∀j ,
+
∆j W (t)(v 0 , v + 1 ) − v + 2
+
∆j∂t W (t)(v 0 , v + 1 ) − v
−→ 0 2
∆j
−
W (t)(v0 , v − 1 ) − v + 2
∆j∂t
−
W (t)(v0 , v − 1 ) − v −→ 0 2
as t → +∞ ,
where we denote by W (t)(f, g) the free solution of the wave equation associated to the
initial data (f, g).
In general, no classical scattering result for the norm of ˙ B n/2−1
2,∞
5
× ˙ B n/2−2
2,∞
holds.