Schrödinger Flow near Harmonic Maps
S. GUSTAFSON
University of British Columbia
K. KANG
Sungkyunkwan University
AND
T.-P. TSAI
University of British Columbia
Abstract
For the Schrödinger flow from R2 × R+ to the 2-sphere S2 , it is not known if
finite energy solutions can blow up in finite time. We study equivariant solutions
whose energy is near the energy of the family of equivariant harmonic maps. We
prove that such solutions remain close to the harmonic maps until the blowup
time (if any), and that they blow up if and only if the length scale of the nearest
c 2006 Wiley Periodicals, Inc.
harmonic map goes to 0. 1 Introduction and Main Result
The Schrödinger flow for maps from Rn to S2 (also known as the Schrödinger
map, and, in ferromagnetism, as the Heisenberg model or Landau-Lifshitz equation) is given by the equation
(1.1)
u t = u × u,
u(x, 0) = u 0 (x).
Here u(x, t) is the unknown map from Rn × R+ to the 2-sphere
S2 := {u ∈ R3 | |u| = 1} ⊂ R3 ,
denotes the Laplace operator in Rn , and × denotes the cross product in R3 . A
more geometric way to write this equation is
(1.2)
u t = J Pu,
Pu = u + |∇u|2 u,
where P = Pu denotes the orthogonal projection from R3 onto the tangent plane
Tu S2 := {ξ ∈ R3 | ξ · u = 0}
to S2 at u, and J = J u := u × is a rotation through π/2 on Tu S2 .
On one hand, equation (1.1) is a borderline case of the Landau-Lifshitz-Gilbert
equations that model isotropic ferromagnetic spin systems:
(1.3)
u t = a Pu + b J Pu,
a ≥ 0,
Communications on Pure and Applied Mathematics, Vol. LX, 0463–0499 (2007)
c 2006 Wiley Periodicals, Inc.
464
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
(see, e.g., [10, 11]). The Schrödinger flow corresponds to the case a = 0. The case
b = 0 is the well-studied harmonic map heat flow, for which some finite-energy
solutions do blow up in finite time [2].
On the other hand, equation (1.1) is a particular case of the Schrödinger flow
from a Riemannian manifold into another one with a complex structure (see, e.g.,
[6, 7, 8, 9, 14, 18, 19]). We will limit ourselves to the case u : Rn × R+ → S2 in
this paper.
Equation (1.1) can be written in the divergence form u t = nj=1 ∂ j (u × ∂ j u),
which is useful in the construction of global weak solutions [16]. Its formal equivalence to a nonlinear Schrödinger equation (NLS) can be seen by applying the
stereographic projection from S2 to C∞ , the extended complex plane,
(1.4)
w=
u 1 + iu 2
,
1 + u3
iwt = −w +
2w̄ (∂ j w)2 .
1 + |w|2 j
It is also known to be equivalent to an integrable cubic NLS in space dimension
n = 1 (see, e.g., [5, 17]).
Equation (1.1) formally conserves the energy
(1.5)
1
E(u) =
2
1
|∇u| d x =
2
2
Rn
3
n |∂ j u k |2 d x.
Rn j=1 k=1
The space dimension n = 2 is critical in the sense that E(u) is invariant under
scaling. In general,
x
, s > 0.
(1.6)
E(u) = s 2−n E(u s ), u s (x) := u
s
For our problem u : Rn × R+ → S2 , local-in-time well-posedness (LWP) is
established in [16] in the class |u| = 1 and ∇u ∈ H k (Rn ), where k > n/2 + 1
is an integer. The authors there also proved global-in-time well-posedness (GWP)
in the same class when n = 1 and when n ≥ 2 for data that is small in certain
Sobolev norms. For n = 2, global existence is proved in [3] for small energy
radial or equivariant data. Also, for n = 2, LWP for a closely related system of
nonlinear Schrödinger equations is established in [12, 13] for data corresponding
to ∇u ∈ H 1+ . There are known to be self-similar blowup solutions for n = 2 [4];
however, these do not have finite energy.
Fix m ∈ Z, a nonzero integer. By an m-equivariant map u : R2 → S2 , we mean
a map of the form
(1.7)
u(r, θ) = emθ R v(r )
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
465
where (r, θ) are polar coordinates on R2 , v : [0, ∞) → S2 , and R is the matrix
generating rotations around the u 3 -axis:




0 −1
0
cos α − sin α 0
0
0 , eα R =  sin α
cos α 0 .
(1.8)
R = 1
0
0
0
0
0
1
Radial maps arise as the case m = 0. The class of m-equivariant maps is formally
preserved by the Schrödinger flow.
If u is m-equivariant, we have |∇u|2 = |u r |2 +r −2 |u θ |2 = |vr |2 +(m 2 /r 2 )|Rv|2 ,
and so
∞
m2
(1.9)
E(u) = π
|vr |2 + 2 (v12 + v22 ) r dr.
r
0
If E(u) < ∞, the limits limr →0 v(r ) and limr →∞ v(r ) make sense (see (2.2) in the
next section), and so we must have v(0), v(∞) = ±k̂, where k̂ = (0, 0, 1)T . We
may and will fix v(0) = −k̂. The two cases v(∞) = ±k̂ correspond to different
topological classes of maps. We denote by m the class of m-equivariant maps
with v(∞) = k̂:
(1.10)
m = {u : R2 → S2 | u = emθ R v(r ), E(u) < ∞,
v(0) = −k̂, v(∞) = k̂}.
The energy E(u) can be rewritten as follows:
∞
m2 v
2
2
|vr | + 2 |J Rv| r dr
E(u) = π
r
0
(1.11)
∞
|m| v 2
=π
vr − r J Rv r dr + Emin
0
where J v := v ×, and
∞
Emin = 2π
(1.12)
0
vr ·
∞
= 2π|m|
|m| v
J Rv r dr
r
(v3 )r dr = 2π|m|[v3 (∞) + 1]
0
(using v12 + v22 + v32 = 1). The number Emin , which depends only on the boundary
conditions, is in fact 4π times the absolute value of the degree of the map u, considered as a map from S2 to itself (defined, for example, by integrating the pullback
by u of the volume form on S2 ). It provides a lower bound for the energy of an
m-equivariant map, E(u) ≥ Emin , and this lower bound is attained if and only if
(1.13)
vr =
|m| v
J Rv.
r
466
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
If v(∞) = −k̂, the minimal energy is Emin = 0 and is attained by the constant map
u ≡ −k̂. On the other hand, if v(∞) = k̂, the minimal energy is
E(u) ≥ Emin = 4π|m|
and is attained by the 2-parameter family of harmonic maps
Om := {emθ R h s,α (r ) | s > 0, α ∈ [0, 2π)}
(1.14)
where
h
and
(1.15)

h 1 (r )
h(r ) =  0  ,
h 3 (r )
s,α
(r ) := e
αR
r
,
h
s

h 1 (r ) =
2
,
r |m| + r −|m|
h 3 (r ) =
r |m| − r −|m|
.
r |m| + r −|m|
The fact that h(r ) satisfies (1.13) means
m
m
(1.16)
(h 1 )r = − h 1 h 3 , (h 3 )r = h 21 .
r
r
mθ R
So Om is the orbit of the single harmonic map e
h(r ) under the symmetries of
the energy E that preserve equivariance: scaling and rotation. Explicitly,


cos(mθ + α)h 1 (r/s)
(1.17)
emθ R h s,α (r ) =  sin(mθ + α)h 1 (r/s)  .
h 3 (r/s)
The solution (1.15) is easily found by solving system (1.13) of ODEs directly. Alternately, under the stereographic projection (1.4), equation (1.13) amounts to the
Cauchy-Riemann equations, and these harmonic maps correspond to the antimeromorphic (if m > 0) functions
−m
z̄
iα
, z = r eiθ .
(1.18)
w=e
s
We are now ready to state our main result. We denote u Ḣ k = ∇ k u L 2 .
T HEOREM 1.1 There exist δ > 0 and C0 , C1 > 0 such that if u ∈ C([0, T ); Ḣ 2 ∩
m ) is a solution of the Schrödinger flow (1.1) conserving energy and satisfying
δ12 := E(u 0 ) − 4π|m| < δ 2 ,
then there exist s(t) ∈ C([0, T ); (0, ∞)) and α(t) ∈ C([0, T ); R) so that
r (mθ+α(t))R
(1.19)
h
< C0 δ1 ∀t ∈ [0, T ).
u(x, t) − e
s(t) Ḣ1 (R2 )
Moreover, s(t) > C1 /u(t) Ḣ2 . Furthermore, if T < ∞ is the maximal time of
existence for u in Ḣ 2 (i.e., limt→T − u(t) Ḣ 2 = ∞), then
(1.20)
s(t) = 0.
lim inf
−
t→T
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
467
Remark 1.2.
(1) This theorem can be viewed, on one hand, as an orbital stability result for
the family of harmonic maps (at least up to the possible blowup time), and on the
other hand as a characterization of blowup for energy near Emin : solutions blow up
if and only if the Ḣ 1 -nearest harmonic map “collapses” (i.e., its length scale goes
to 0).
(2) The assumption E(u 0 ) − 4π|m| < δ 2 implies u 0 is close to Om in Ḣ 1
(see (2.7)), but not necessarily in Ḣ 2 .
(3) The existence of local (in time) Ḣ 2 ∩ Ḣ 1 solutions of (1.1) is established
in [16] for sufficiently regular initial data. In particular, this ensures Theorem 1.1
is nonempty (see also [12]–[13] for local well-posedness results). Local wellposedness with data in Ḣ 2 ∩ Ḣ 1 still appears to be open. If we had this, (1.20)
could be replaced by limt→T − s(t) = 0.
(4) From now on we will assume m > 0. The cases m < 0 of Theorem 1.1
follow from the change of variables (x1 , x2 , x3 ) → (x1 , −x2 , x3 ).
The plan for the paper is as follows: In Section 2 we study maps whose energy
is close to that of the family of harmonic maps. The analysis here is completely
time independent. In Section 3 we apply Strichartz estimates to a certain nonlinear Schrödinger equation, obtained via the Hasimoto transformation introduced in
[3], and present the proof of the main theorem. The proofs of some of the more
technical lemmas are relegated to Section 4 in order to streamline the presentation.
Without loss of generality, we assume m > 0 for the rest of the paper.
R EMARK ON N OTATION. Throughout the paper, the letter C is used to denote
a generic constant, the value of which may change from line to line.
2 Maps with Energy near the Harmonic Map Energy
This section is devoted entirely to static m-equivariant maps (i.e., there is no
time dependence anywhere in this section). We establish some properties of maps
with energy close to the harmonic map energy 4πm. Roughly speaking, we prove
that such maps are Ḣ 1 -close to harmonic maps. Precise statements appear in Theorem 2.1 below.
We define the distance from any map u to the family Om of m-equivariant harmonic maps to be
dist(u, Om ) := inf u − emθ R h s,α Ḣ 1 .
s∈(0,∞)
α∈S1
Here S1 = R/2π.
The following theorem defines a (nonlinear) projection from the set m of mequivariant maps with energy close to 4πm onto the family Om and establishes a
key fact: for maps in this set, the squared distance dist2 (u, Om ) is bounded by the
energy difference E(u) − 4πm.
468
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
T HEOREM 2.1 There are constants δ > 0 and C0 , C1 > 0 such that if u ∈ m
satisfies
E(u) < 4πm + δ 2 ,
then the following hold:
(i) There exist unique s(u) ∈ (0, ∞) and α(u) ∈ S1 such that
dist(u, Om ) = u − emθ R h s(u),α(u) Ḣ 1 .
Moreover, s(u) and α(u) are continuous functions of u ∈ Ḣ 1 .
(ii) dist(u, Om ) < C0 [E(u) − 4πm]1/2 < C0 δ.
(iii) If u ∈ Ḣ 2 (R2 ), then s(u)u Ḣ 2 (R2 ) > C1 .
P ROOF : The proof is long, so we break it into a series of steps. At each step,
we may need to take δ smaller than in the previous step.
Step 1. A Change of Variable. Recall that u ∈ m implies that in polar
coordinates (r, θ), u(x) = emθ R v(r ), with v(0) = −k̂ and v(∞) = k̂. The change
of variables
r → y = m log(r ) ∈ (−∞, ∞) or e y = r m
turns out to be very useful for our purposes.
Set
ṽ(y) := v(e y/m ).
With this change of variables, the Ḣ 1 inner product of m-equivariant maps changes
as follows:
mθ R
mθ R
e v(r ), e w(r ) Ḣ 1 = ∇[emθ R v(r )] · ∇[emθ R w(r )]d x
R2
m2
vr · wr + 2 Rv · Rw r dr
r
∞
= 2π
0
= 2πm
(ṽ · w̃ + R ṽ · R w̃)dy,
R
where denotes d/dy. In particular,
(2.1)
1
Ẽ(ṽ) :=
2
E(u) = 2πm Ẽ(ṽ),
|ṽ (y)|2 + ṽ1 (y)2 + ṽ2 (y)2 dy.
R
Note that this implies ṽ j ∈ H (R) for j = 1, 2, and in particular (ṽ j2 ) ∈ L 1 (R) so
that the limits lim y→±∞ ṽ j2 (y) exist and are equal 0. The function ṽ is continuous,
and ṽ32 = 1 − ṽ12 − ṽ22 has limit 1 as y → ±∞. Thus the limits lim y→±∞ ṽ exist,
justifying our earlier claim
1
(2.2)
lim v(r ) and lim v(r ) exist.
r →0
r →∞
Recall that for v ∈ m , we have chosen ṽ(−∞) = −k̂, ṽ(∞) = k̂.
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
469
Ẽ(ṽ) inherits the “topological lower bound”
1
(2.3)
Ẽ(ṽ) = 2 +
|ṽ − J ṽ R ṽ|2 dy ≥ 2.
2
R
In this new variable, scaling r corresponds to translating y. In particular, the family
of harmonic maps is composed of translations and rotations of a simple explicit
map:
r
s,α
αR
= eα R h̃(y − m log(s)),
h (r ) = e h
s


sech y
h̃(y) =  0  .
tanh y
Step 2. Energy Close to 4πm Implies u Close to a Harmonic Map. Using
the variable y, we would like to prove the following:
L EMMA 2.2 For any > 0, there exists µ > 0 such that if a map ṽ : R → S2
satisfies Ẽ(ṽ) < 2 + µ, and ṽ(−∞) = −k̂, ṽ(∞) = k̂, then
inf
α∈S1 ,a∈R
ṽ − eα R h̃( · − a) H 1 (R) < .
P ROOF : Suppose not. Then there exist v j (r ), j = 1, 2, 3, . . . , and 0 > 0 such
that
|v j (r )| ≡ 1,
(2.4)
v j (±∞) = ±k̂,
1
Ẽ(v j ) < 2 + ,
j
v j − eα R h̃( · − a) H 1 (R) ≥ 0 ,
for every α ∈ S1 and a ∈ R. Since Ẽ(v j ) < ∞, v j is continuous and thus
v j3 (a j ) = 0 for some a j ∈ R. We replace v j by w j (y) := e−αj R v j (y + a j ),
where α j ∈ S1 is chosen so that w j (0) = ı̂ = (1, 0, 0)T . The properties (2.4) still
hold for {w j }:
(2.5) |w j (r )| ≡ 1,
1
Ẽ(w j ) < 2 + ,
j
w j (±∞) = ±k̂,
w j − h̃ H 1 (R) ≥ 0 .
Since sup j Ẽ(w j ) < ∞, there is a subsequence (which we continue to denote by
{w j }) and a limit vector function w∗ (y) satisfying

∗
∗
1

w j1 , −→ w1 , w j2 −→ w2 weakly in H (R)
(2.6)
weakly in L 2 (R)
wj3 −→ w3∗


0
strongly in L 2loc (R) ∩ Cloc
(R).
w j −→ w∗
470
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Because of the local uniform convergence, we have |w∗ | ≡ 1 and w∗ (0) = ı̂. On
the other hand, by the topological lower bound (2.3), we have
1
1
|wj − J wj Rw j |2 < 2 + ,
2 ≤ Ẽ(w j ) = 2 +
2
j
R
w j
− J wj Rw j L 2 (R) → 0 as j → ∞. For any
from which it is immediate that
bounded interval I , using J w Rw = k̂ − w3 w, we have
∗
J wj Rw j − J w Rw∗ L 2 (I ) = w j3 w j − w3∗ w∗ L 2 (I ) → 0 as j → ∞.
∗
Hence w j → J w Rw∗ strongly in L 2 (I ). On the other hand, w∗ is the weak limit
∗
of wj , and so we obtain w∗ = J w Rw∗ almost everywhere, with w∗ (0) = ı̂. By
1
uniqueness of Hloc
solutions of this system of ordinary differential equations, we
must have
w∗ (y) = h̃(y) = ( sech y, 0, tanh y)T .
Now we note that
Ẽ(w∗ ) = Ẽ(h̃) = 2 = lim Ẽ(w j ).
j→∞
By (2.1),
Rw∗ 2H 1 (R) + w3∗ 2L 2 (R) = lim Rw j 2H 1 (R) + wj3 2L 2 (R) .
j→∞
By weak lower semicontinuity, i.e.,
Rw∗ 2H 1 (R) ≤ lim inf Rw j 2H 1 (R) ,
j→0
w3∗ 2L 2 (R) ≤ lim inf wj3 2L 2 (R) ,
j→0
we have Rw j 2H 1 (R) → Rw∗ 2H 1 (R) and wj3 2L 2 (R) → w3∗ 2L 2 (R) , which im
plies Rw j → Rw∗ strongly in H 1 (R) and wj3 → w3∗ strongly in L 2 (R). Finally,
we will show that w j − h̃ converges to 0 strongly in H 1 (R), which will contradict
assumption (2.5), and so complete the proof of the lemma. Indeed,
w j − h̃2H 1 (R) = wj − h̃ 2L 2 (R) + Rw j − R h̃2L 2 (R) + w j3 − h̃ 3 2L 2 (R) .
We have already shown that the first two terms go to 0 in the limit, and so it remains
to consider the last term. For this, we need another lemma. For f : (a, b) → R,
denote by T(a,b) ( f ) the total variation of f on (a, b). The following lemma shows
that the total variation of v3 is close to 2 if E(u) is close to 4πm.
L EMMA 2.3 If u = emθ R v(r ) ∈ m and E(u) = 4πm + 0 , then T(0,∞) (v3 ) ≤
2 + C0 .
P ROOF : Make the change of variable ṽ(y) = v(e y/m ), and write
ṽ = (ρ̃ cos(ω̃), ρ̃ sin(ω̃), ṽ3 ),
so that ρ̃ =
2
ṽ12
+
ṽ22 .
We have
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
0
4+
≥
πm
R
471
(ρ̃ )2
2
(|ṽ | + |R ṽ| )dy =
+ ρ̃ dy
ρ̃ (ω̃ ) +
1 − ρ̃ 2
R
d
ρ̃|ρ̃ |
2 1/2 dy
=
2
)
(1
−
ρ̃
≥2
dy
dy = 2TR (ṽ3 ).
(1 − ρ̃ 2 )1/2
2
2
2
2
R
R
Dividing by 2 on both sides completes the proof.
Applying this lemma to w j , we have TR (w j3 ) ≤ 2 + C/j. Since w j3 (−∞) =
−1, w j3 (0) = 0, and w j3 (∞) = 1, we have w j3 (y) > −C/j for y ≥ 0 (and
similarly for y ≤ 0). Fix 1 > 0. For |y| ≥ 1 and j sufficiently large (depending
on 1 ),
|w3 j − h̃ 3 | =
|w3 j + h̃ 3 |
≤
|w32 j − h̃ 23 |
|h̃ 3 | − C/j
2 2
2 ≤
w3 j − h̃ 23 =
|R h̃|2 − |Rw j |2 ,
tanh 1
tanh 1
and so
Since
|w32 j − h̃ 23 |
|y|≥1
|y|<1
|w3 j − h̃ 3 |2 dy → 0 as j → ∞.
|w3 j − h̃ 3 |2 dy ≤ C1 and 1 is arbitrary, we conclude
w3 j − h̃ 3 L 2 (R) → 0.
This completes the proof of Lemma 2.2.
Translating Lemma 2.2 back to the original variable r = e y/m , we find
given > 0, there is µ > 0 such that if u ∈ m , E(u) < 4πm + µ,
(2.7)
then dist(u, Om ) < .
Step 3. Existence of s(u) and α(u). Recall that since u ∈ m , E(u) ≥
4πm, and set δ1 := [E(u) − 4πm]1/2 < δ. We observe first that
(2.8)
lim inf u − emθ R h s,α 2Ḣ 1 = lim inf u − emθ R h s,α 2Ḣ 1 = 8πm + δ12 .
s→∞ α
s→0 α
Indeed, we have
(2.9)
u −
emθ R h s,α 2Ḣ 1
= 8πm +
δ12
−2
∇u · ∇(emθ R h s,α )d x,
R2
and it suffices to show that for any α ∈ S1
(2.10)
∇u · ∇(emθ R h s,α )d x → 0
R2
as s → 0 or s → ∞. Since h s,α (r ) = h 1,α (r/s), this latter fact follows from an
easy lemma:
472
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
L EMMA 2.4 If f ∈ L 2 (R2 ), then
1
x
→ 0 weakly in L 2 (R2 ) as s → ∞ and s → 0.
f
s
s
P ROOF : First suppose f ∈ L 2 ∩ L ∞ and fix g ∈ L 2 . By Hölder’s inequality,
1
1
x
x
x
1
f
g(x)d x =
f
g(x)d x +
f
g(x)d x
s
s
s
s
s
s
√
√
R2
|x|≤ s
|x|> s
√
1/2
π
2
|g(x)| d x
→0
≤ f L ∞ g L 2 √ + f L 2
s
√
|x|> s
as s → ∞. A similar argument covers the s → 0 case. For general f ∈ L 2 , choose
f ∈ L 2 ∩ L ∞ with f − f L 2 < . Then by the above argument,
1
x
≤ g L 2 + o(s).
f
g(x)d
x
s
s
R2
Since is arbitrary, we are done.
To prove the claim (2.10), just take g = ∇u and 1s f ( · /s) = ∇(emθ R h s,α ) in
this lemma.
So on one hand, infα u − emθ R h s,α 2Ḣ 1 approaches 8πm + δ12 as s → 0 and
s → ∞. On the other hand, (2.7) shows that if δ (and hence δ1 ) is sufficiently
small, then for some s and α, u − emθ R h s,α 2Ḣ 1 < 8πm. Thus to minimize
F(s, α) := u − emθ R h s,α 2Ḣ 1
over s ∈ (0, ∞), α ∈ S1 , it suffices to consider s in a compact subset of (0, ∞).
Since F(s, α) is continuous, there must exist s(u) ∈ (0, ∞) and α(u) ∈ S1 such
that
dist(u, Om ) = u − emθ R h s(u),α(u) Ḣ 1 .
Step 4. Uniqueness of s(u) and α(u). Denote σ = (s, α). Suppose there
exist σ1 and σ2 with σ1 = σ2 such that
δ0 := dist(u, Om ) = u − emθ R h σ1 Ḣ 1 = u − emθ R h σ2 Ḣ 1 .
Let µ be half the distance between emθ R h σ1 and emθ R h σ2 , µ := 12 emθ R h σ1 −
emθ R h σ2 Ḣ 1 . It follows that µ ≤ δ0 . Now set ϕ(t) = emθ R h σ (t) with σ (t) =
1
[(σ1 + σ2 ) + t (σ2 − σ1 )], so that ϕ(−1) = emθ R h σ1 and ϕ(1) = emθ R h σ2 . Set
2
ϕ̄ := 12 [ϕ(−1) + ϕ(1)]. Lemma 4.6 (stated and proved in Section 4) then yields
ϕ̄ − ϕ(0) Ḣ 1 < Cµ2 .
This estimate amounts to a bound on the curvature of the family Om .
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
473
Now ∇[u− ϕ̄] is L 2 -orthogonal to ∇[emθ R (h σ1 −h σ2 )], since u−emθ R h σ1 Ḣ 1 =
u − emθ R h σ2 Ḣ 1 . By the Pythagorean theorem,
1
u − ϕ̄2Ḣ 1 = u − ϕ(−1)2Ḣ 1 − ϕ(−1) − ϕ(1)2Ḣ 1 = δ02 − µ2 ,
4
and so
u − ϕ(0) Ḣ 1 ≤ u − ϕ̄ Ḣ 1 + ϕ̄ − ϕ(0) Ḣ 1 ≤ (δ02 − µ2 )1/2 + Cµ2
= δ0 −
µ2
+ Cµ2 .
2
2
δ0 + δ0 − µ
By (2.7), we can ensure δ0 < 1/(2C) by choosing δ sufficiently small. This in
turn implies u − ϕ(0) Ḣ 1 < δ0 , which contradicts the assumption that δ0 is the
minimal distance. This establishes uniqueness of s(u) and α(u).
Step 5. Continuity of s(u) and α(u). We could invoke the implicit function
theorem, but we prefer to give a simple direct proof of continuity. Suppose u j → u
in Ḣ 1 with E(u j ) < 4πm + δ 2 and E(u) < 4πm + δ 2 . We have
dist(u j , Om ) ≤ u j − emθ R h s(u),α(u) Ḣ 1 ≤ u j − u Ḣ 1 + dist(u, Om )
and
dist(u, Om ) ≤ u − emθ R h s(u j ),α(u j ) Ḣ 1 ≤ u j − u Ḣ 1 + dist(u j , Om )
and so dist(u j , Om ) → dist(u ∗ , Om ). Since
emθ R (h s(u j ),α(u j ) − h s(u),α(u) ) Ḣ 1 ≤ dist(u j , Om ) + dist(u, Om ) + u j − u Ḣ 1 ,
{s(u j )} is contained in a compact subinterval of (0, ∞) by Lemma 4.5, and so,
up to subsequence, s(u j ) → s∗ and α(u j ) → α∗ for some s∗ and α∗ . Along this
subsequence
u − h s(u),α(u) Ḣ 1 = dist(u, Om ) = lim dist(u j , Om )
j→∞
= lim u j − h s(u j ),α(u j ) Ḣ 1 = u − h s∗ ,α∗ Ḣ 1 .
j→∞
By the uniqueness we have already proved, s∗ = s(u) and α∗ = α(u). We conclude
that s(u j ) → s(u) and α(u j ) → α(u) (for the full sequence). Continuity is proved.
This completes the proof of part (i) of Theorem 2.1.
Step 6. The “Linearized Operator.” We now proceed to the proof of part (ii)
of Theorem 2.1. The main idea is this: the “global” result of Lemma 2.2 allows
us now to work “locally” — i.e., nearby a harmonic map. Indeed, to prove (ii),
we study the second variation of the energy functional around the nearby harmonic
map. We begin by discussing this “linearized operator.”
Given an m-equivariant map u ∈ m with E(u) − 4πm < δ 2 , we fix s = s(u)
and α = α(u) and write u = emθ R v(r ) with
r
r
αR
+ξ
.
(2.11)
v(r ) = e
h
s
s
474
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
This defines ξ = ξ(r ) = e−α R v(sr ) − h(r ) (note that the variable r here is no
longer the original polar coordinate). Using h r − (m/r )J h Rh = 0, expand
1 m v 2
1
2
|∇u| d x = 4πm +
E(u) =
vr − J Rv d x
2
2 r
R2
1
= 4πm +
2
(2.12)
2
Lξ + m ξ3 ξ d x
r
R2
where Lξ is the linear part,
m
(ξ3 h + h 3 ξ ).
r
In fact, the operator L maps tangent vector fields (i.e., vector functions η(r ) tangent to S2 at h(r )) into tangent vector fields. To see this explicitly, we specify an
orthonormal basis of Th S2 :
 


0
−h 3 (r )
e = 1 and J h e =  0  .
h 1 (r )
0
(2.13)
Lξ := ξr +
Then for any map ξ : [0, ∞) → R3 , we have an orthogonal decomposition
(2.14)
ξ(r ) = z 1 (r )e + z 2 (r )J h e + γ (r )h,
which defines a complex-valued function z(r ) := z 1 (r ) + i z 2 (r ). Note that
m
m
m
m
Le = h 3 e, L(J h e) = h 3 J h e, Lh = h 3 h + k̂.
r
r
r
r
2
Hence the operator L restricted to Th S is equivalent to
m
1
L 0 := ∂r + h 3 = h 1 ∂r
r
h1
in the sense that
L(z 1 e + z 2 J h e) = (L 0 z 1 )e + (L 0 z 2 )J h e.
Roughly speaking, our strategy for proving part (ii) of Theorem 2.1 is to show
that dist(u, Om ) is controlled by z, which is controlled by L 0 z, which in turn is
controlled by E(u) − 4πm.
Define for radial complex-valued functions f (r ) and g(r ) the following inner
product:
∞
m2 ¯
¯
fr (r )gr (r ) + 2 f (r )g(r ) r dr.
f, g X :=
r
0
y/m
y/m
If we set f˜(y) := f (e ) and g̃(y) := g(e ), we have
f˜ g̃ + f˜g̃ dy = m f˜, g̃ H 1 (R) .
(2.15)
f, g X = m
R
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
475
L EMMA 2.5 There are ε, C > 0 such that if f : [0, ∞) → C satisfies
| f, h 1 X | ≤ ε f X
(2.16)
then
(2.17)
f
2X
∞
=
0
∞
m2 2
2
| fr | + 2 | f | r dr ≤ C
|L 0 f |2r dr.
r
0
P ROOF : We may assume f (r ) is real-valued. Under our change of variables
f˜(y) := f (e y/m ), we have
∞
m2
fr2 + 2 f 2 r dr = m (( f˜ )2 + f˜2 )dy
r
0
R
and
∞
[L 0 f ] r dr = m
2
0
[ L̃ 0 f˜]2 dy
R
where
1
d
d
+ tanh y = sech (y)
.
dy
dy sech y
In the variable y, the assumption of the lemma becomes
ε
f˜ ( sech y) + f˜ sech y dy ≤ √ f˜ H 1 (R) ,
(2.18)
f˜, sech y H 1 (R) =
m
L̃ 0 :=
R
and so it suffices to prove that (2.18) implies
(2.19)
f˜2H 1 (R) ≤ C L̃ 0 f˜2L 2 (R) = C f˜, L̃ ∗0 L̃ 0 f˜ L 2 (R) .
The second-order differential operator
d2
+ 1 − 2 sech 2 (y)
dy 2
is nonnegative with unique zero eigenfunction (“ground state”)√sech (y) (in fact,
this operator is well studied; see, e.g., [15]). Set φ := (1/ 2) sech y so that
φ L 2 (R) = 1. Write
H := L̃ ∗0 L̃ 0 = −
f˜ = aφ + ϕ
with φ, ϕ = 0.
Similarly, decompose
φ = b(φ − φ) + ψ with (φ − φ), ψ = 0.
√
Since φ − φ = − 2 sech 3 y, ψ is nonzero, b = −φ 2 2L 2 (R) /(2φ 3 2L 2 (R) ), and
ψ L 2 < 1.
Now by (2.18),
|a| = |φ, f˜| ≤ |b(φ − φ), f˜| + |ψ, f˜|
= |b| |φ, f˜ H 1 | + |ψ, f˜| ≤ Cε f˜ H 1 (R) + |ψ, f˜|.
476
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Using the above estimate and (A + B)2 ≤ q A2 + q B 2 where q is the Hölder
conjugate of q with 1 < q < ∞, we have
f˜, f˜ = |a|2 + ϕ2L 2 ≤ (Cε f˜ H 1 (R) + |ψ, f˜|)2 + ϕ2L 2
≤ Cq ε2 f˜2H 1 (R) + q|ψ, f˜|2 + ϕ2L 2
≤ Cq ε2 f˜2H 1 (R) + qψ2L 2 f˜2L 2 + ϕ2L 2 .
Choose q to be such that qψ2L 2 < 1 to obtain
f˜2L 2 ≤ Cε2 f˜2H 1 (R) + C f˜, H f˜,
where we used ϕ2L 2 ≤ Cϕ, H ϕ = C f˜, H f˜. On the other hand, we have
f˜ , f˜ ≤ f˜, H f˜) + C f˜2L 2 .
Combining the two estimates above, we get
f˜2H 1 (R) ≤ C f˜, H f˜ = C
[ L̃ 0 f˜]2 dy,
provided ε is sufficiently small. Transforming back to the variable r , we obtain
estimate (2.17), completing the proof.
Step 7. Almost Orthogonality. To apply the previous lemma to z(r ), we
need to verify condition (2.16), for which we use the following lemma:
L EMMA 2.6 For z(r ) and γ (r ) defined by (2.11) and (2.14),
∞
m2
(2.20)
z 1r h 1r + 2 z 1 h 1 r dr = 0,
z 1 , h 1 X =
r
0
∞
∞
m2
4m 2 2
z 2r h 1r + 2 z 2 h 1 r dr =
(2.21)
h h 3 γ r dr.
z 2 , h 1 X =
r
r2 1
0
0
P ROOF : The pair (α(u), s(u)) is the minimizer of the differentiable function
2
F(s, α) = ∇(emθ R (v − h α,s ))
d x.
R2
The lemma follows from the equations ∇α,s F(α(u), s(u)) = 0.
Step 8. Proof of (ii). We will use the following abbreviation:
ze := z 1 e + z 2 J h e,
z = z1 + i z2.
By (2.7), we may choose δ sufficiently small so that
u − emθ R h s,α Ḣ 1 < δ0
for any given δ0 > 0 (which will be specified later). So we have
2
m
2
mθ R s,α 2
(2.22)
ξr L 2 (R2 ) + h Ḣ 1 < δ02 .
r Rξ 2 2 = u − e
L (R )
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
477
It is proved in Lemma 4.7 that (2.22) implies the L ∞ -smallness of ξ(r ) for δ0
sufficiently small: ξ L ∞ ≤ Cδ0 . This immediately implies
z∞ + γ ∞ ≤ Cδ0 .
(2.23)
Since 1 = |v|2 = |z|2 + (1 + γ )2 and |γ | is small,
(2.24) γ = 1 − |z|2 − 1 ≤ 0, |γ (r )| ≤ C|z(r )|2 ,
|γr (r )| ≤ C|z(r )zr (r )|.
By Lemma 2.6, equation (2.23) through (2.24), and |h(r )| ≤ 1,
∞
h1 z 4m 2 2
≤ Cδ0 z X .
|z, h 1 X | = h
h
γ
r
dr
≤
Cz
3
∞
1
r r r2
0
2
2
Taking δ0 small enough so that Cδ0 is less than ε in Lemma 2.5, we have
∞
m2
2
2
2
dist (u, Om ) = 2π
|ξr | + 2 |Rξ | r dr
r
0
(2.25)
∞
∞
|z|2
2
|zr | + 2 r dr ≤ C
|L 0 z|2r dr.
≤C
r
0
0
On the other hand, by (2.12),
2
1 1
m
2
δ1 = E(u) − 4πm =
Lξ + ξ3 ξ d x =
2 r
2
R2
Hence
R2
|L 0 z| d x =
|L(ze)| d x ≤
2
R2
2
L(ze) + L(γ h) + m ξ3 ξ d x.
r
2
3δ12
2 m
2
+C
|L(γ h)| + ξ3 ξ d x.
r
R2
R2
Using (2.23) we have, for δ0 sufficiently small,
2
m
ξ3 ξ d x ≤ Cz2
∞
r
R2
2
z
d x.
r R2
For the term L(γ h)2L 2 , using (2.24) we find
2 2 γ h γ
2
2
|L(γ h)| ≤ C
|γr h| + + h 3 h ≤ Cz∞ |L 0 z|2 .
r
r
2
R2
R2
R2
Thus we get (1 − Cδ02 )L 0 z2L 2 ≤ 4δ12 . Now choose δ0 > 0 sufficiently small (by
choosing δ small) so that L 0 z2L 2 ≤ 5δ12 and therefore dist(u, Om ) < Cδ1 . This
completes the proof of part (ii) of Theorem 2.1.
478
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Step 9. Proof of (iii). By (2.9) with s = s(u) and α = α(u), we have
mθ R s,α
h )d x > 2πm, since δ1 is small. On the other hand, inequalR2 ∇u · ∇(e
ity (4.2) from Lemma 4.1 in Section 4 gives an upper bound: for any σ ∈ (0, 1),
1/2 1/2
|∇u|2
2σ
mθ R s,α 2
∇u · ∇(emθ R h s,α )d x ≤
d
x
|x|
|∇(e
h
)|
d
x
|x|2σ
R2
R2
R2
≤ C∇u1−σ
∇ 2 uσL 2 (R2 ) |x|σ |∇(emθ R h s,α )| L 2 (R2 )
L 2 (R2 )
≤ C∇ 2 uσL 2 (R2 ) s σ .
Thus we obtain ∇ 2 u L 2 (R2 ) ≥ C/s, completing the proof of Theorem 2.1.
3 Global Well-Posedness vs. Blowup
In this section we complete the proof of our main result, Theorem 1.1.
Let u ∈ C([0, T ); Ḣ 2 ∩m ) be a solution of the Schrödinger flow equation (1.1)
that conserves energy. We are assuming
δ12 := E(u 0 ) − 4πm = E(u(t)) − 4πm < δ 2
where δ is taken to be sufficiently small. In particular, we choose δ small enough
so that Theorem 2.1 applies for each t ∈ [0, T ) and so furnishes us with continuous
functions s(t) ∈ (0, ∞) and α(t) ∈ R such that
r (mθ+α(t))R
< Cδ1
h
(3.1)
u(x, t) − e
s(t) 1
Ḣ
and
s(t)u(t) Ḣ 2 > C > 0.
Thus the first part of Theorem 1.1 is proved. It remains to show that
(3.2)
T <∞
and
lim u(t) Ḣ 2 = ∞ ⇒ lim inf
s(t) = 0.
−
t→T −
t→T
Recall that we are writing
u(x, t) = e
mθ R
v(r ),
v(r ) = e
and (3.1) is equivalent to
e
mθ R
ξ(r )2Ḣ 1
=
ξr (r )2L 2
α(t)R
r
r
+ξ
,
h
s(t)
s(t)
2
2 Rξ(r ) +m r
L2
< Cδ12 .
To prove (3.2), we need estimates showing that u(t) Ḣ 2 is controlled as long
as s(t) is bounded away from zero. These Ḣ 2 -estimates are obtained using the fact
that the coordinates of the tangent vector field vr − (m/r )J v Rv with respect to a
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
479
certain orthonormal frame satisfy a nonlinear Schrödinger-type equation and can
be estimated using Strichartz estimates.
This construction, which was introduced in [3], begins with a unit tangent vector field ê(r ) ∈ Tv(r ) S2 satisfying the parallel transport condition
Drv ê ≡ 0.
Recall that Drv is the covariant derivative acting on vector fields η(r ) ∈ Tv(r ) S2 :
Drv η := P v ηr = ηr − (v · ηr )v = ηr + (vr · η)v.
So ê(r ) and J v ê(r ) form an orthonormal frame on Tv S2 . Then q(r ) = q1 (r ) +
iq2 (r ) is defined to be the coordinates of vr − (m/r )J v Rv ∈ Tv S2 in the basis
m
vr − J v Rv = q1 ê + q2 J v ê.
r
We will sometimes write q ê := q1 ê + q2 J v ê for convenience. Note that by (1.11),
2
m
1
δ2
2
v
q L 2 (r dr ) = = (E(u) − 4πm) = 1
vr − r J Rv 2
π
π
L (r dr )
is constant in time and can be taken small.
Define ν = ν1 + iν2 as follows:
J v Rv = ν1 ê + ν2 J v ê.
Again, we will sometimes denote ν ê := ν1 ê + ν2 J v ê. It is now straightforward,
if somewhat involved, to show that if u(x, t) solves the Schrödinger map equation (1.1), the complex function q(r, t) solves the following nonlinear Schrödinger
equation with nonlocal nonlinearity (see [3] for more details):
(1 − mv3 )2
m(v3 )r
q + q N (q),
q+
2
r
r
where
∞
1 − mv3
m
q dr.
q̄ + ν̄ qr +
N (q) = Re
r
r
r
By changing variables to q̃ := ei(m+1)θ q, we obtain
m(1 + v3 )(mv3 − m − 2)
mv3,r
(3.4)
i q̃t + q̃ −
q̃ −
q̃ − q̃ N (q) = 0.
2
r
r
We will use this equation to obtain H 1 -estimates on q.
For these estimates of q to be useful, we need to bound the original map u(x, t)
— or, equivalently, v(r, t) or ξ(r, t) or z(r, t) — by q. Since v(r, t) = eα R [h(r/s)+
ξ(r/s, t)], we have
r
1 αR
m
(3.5)
q ê = e
Lξ + ξ3 ξ
,
s
r
s
where, recall,
m
Lξ = ξr + (ξ3 h + h 3 ξ ) and ξ = z 1 e + z 2 J h e + γ h.
r
(3.3)
iqt = −r q +
480
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Since ξ is small, we have, very roughly speaking,
ξ ≈ ze
Lξ ≈ (L 0 z)e,
and
and z can be controlled by L 0 z. More precisely, Lemma 4.8, proved in the next
section, gives the following bounds: for 2 ≤ p < ∞, and provided δ is sufficiently
small,
z
(3.6) zr L p (R2 ) + r p 2
L (R )
1−2/ p
≤C s
q L p (R2 ) + q L 2 (R2 ) ,
q 2
(3.7) zrr L 2 (R2 ) ≤ C sqr L 2 (R2 ) + s +
sq
+
q
2
2
L (R ) ,
L 4 (R2 )
r 2 2
L (R )
q 1
1
2
+
q
+
+ qr L 2 (R2 ) + q
(3.8) u Ḣ 2 (R2 ) ≤ C
2
2
L (R ) .
L 4 (R2 )
r 2 2
s
s
L (R )
We will use the following notation to denote space-time Lebesgue norms: for
an interval I ⊂ R,
r/ p
r
p
| f (x, t)| d x
dt.
f L r L p (R2 ×I ) :=
t
x
I
R2
We use (3.6)–(3.7) together with Strichartz estimates and equation (3.4) to prove
the following estimates for q:
L EMMA 3.1 For τ ≥ 0 and σ > 0, set I := (τ, τ + σ ), Q := R2 × I , and
8/3 8
2
X (Q) := L 4t L 4x (Q) ∩ L ∞
t L x (Q) ∩ L t L x (Q). Define s := inft∈I s(t). If δ is
sufficiently small, we have
q X (Q) ≤ C q0 L 2x (R2 ) + s−1 σ 1/2 + q2L 4 L 4 (Q) q L 4x L 4t (Q) ,
(3.9)
x t
(3.10) ∇ q̃ X (Q) ≤ C ∇ q̃(τ ) L 2x
+ s−1 σ 1/2 + s−3/2 σ 3/4 + q2X (Q) ∇ q̃ L ∞
2 (Q)∩L 4 L 4 (Q) .
L
t
t x
x
Before proving Lemma 3.1, we show how it completes the proof of our main
theorem.
C OMPLETION
pose that
(3.11)
OF
P ROOF OF T HEOREM 1.1: We need to prove (3.2), so suplim u(t) Ḣ 2 = ∞
t→T −
and
(3.12)
lim inf
s(t) = s0 > 0.
−
t→T
Our goal is to derive a contradiction. By (3.12), we have s(t) ≥ s ∗ for all 0 ≤
t < T for some s ∗ > 0. So we may take s = s ∗ in the estimates (3.9)–(3.10) for
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
481
any time interval I ⊂ [0, T ). If σ is sufficiently small depending on s ∗ (σ 1/2 <
s ∗ /(2C)), and q0 L 2 ≤ Cδ1 is taken sufficiently small, estimate (3.9) implies
q X (Q) ≤ Cδ1 .
Using this estimate in (3.10), for δ1 and σ sufficiently small we obtain
∇ q̃ L ∞
≤ C∇ q̃(τ ) L 2 .
2
t L x (Q)
In particular, taking τ close to T , we see
lim sup ∇ q̃(t) L 2 < ∞.
t→T −
Then using (3.8) and |qr | + |q/r | ≤ C|∇ q̃|, we find lim supt→T − u(t) Ḣ 2 < ∞,
contradicting (3.11). This completes the proof of Theorem 1.1.
P ROOF OF L EMMA 3.1: By applying Strichartz estimates for the inhomogeneous Schrödinger equation (see, e.g., [1]) to (3.4), we get
)
q X (Q) ≤ C(q(τ ) L 2 + F L 4/3 L 4/3
x (Q)
(3.13)
t
where
m(v3 )r
m(1 + v3 )(mv3 − m − 2)
q̃ +
q̃ + q̃ N (q).
2
r
r
Conservation of the L 2 -norm of q(t) (equivalent to conservation of energy E(u))
means we can replace q(τ ) L 2 by q0 L 2 in (3.13). Using v3 (r ) = h 3 (r/s) +
ξ3 (r/s) and ξ3 = z 2 h 1 + γ h 3 , we find
1 + v3 = 1 1 + h3 + z2h1 + γ h3 2
r2 2
s
r2
Lx
L
x
z2h1 γ h3 C 1 + h3 ≤
+
+
s r 2 L 2x r 2 L 2x r 2 L 2x
(3.14)
2 z
z
C
1+ +
≤
s
r L 2x r L 4x
F :=
≤ C(s −1 (1 + q L 2 + q2L 2 ) + q2L 4 ).
Using (3.14) and the uniform boundedness of q L 2 , we get
1 + v3 m(1 + v3 )(mv3 − m − 2) q̃ 4/3 4/3 ≤ C r 2 q̃ 4/3 4/3
r2
Lt Lx
L L
t x 1 + v3 ≤ C
r 2 2 q L 4x Lx
4/3
Lt
≤ C s−1 σ 1/2 + q2L 4 L 4 q L 4t L 4x .
t
x
482
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Next we estimate (m(v3 )r /r )q L 4/3 L 4/3
. Compute
x
t
1
1 mh 21
mh 1 h 3 z 2
mγ h 21 r (v3 )r =
+ h 1 (z 2 )r −
+ γr h 3 +
.
r
sr
r
r
r
s
Remark 3.2. The notation means evaluate the quantity in the square brackets with
r replaced by r/s. We use this notation frequently in what follows.
Therefore, using boundedness of z, h 3 , h 1 /r , and γ = O(|z|2 ), γr = O(|zzr |),
we have
2 (v3 )r z
C h1 −1
(3.15)
≤
+
z
2 + r
L
r 2
r 2 ≤ Cs (1 + q L 2 ).
s r2 2
L
So
m(v3 )r
r
q
4/3 4/3
Lx
Lt
L
L
v3,r −1 1/2
≤ C q L 4x 4/3 ≤ C(1 + q0 L 2 )s σ q L 4t ,L 4x .
r L 2x
Lt
Next we need to estimate the nonlocal term
∞
1 − mv3
m ν̄
qr +
q dr
N (q) = Re
q̄ +
r
r
r
∞
1 − mv3 2 m ν̄
|q|2
m ν̄(1 − mv3 )
=−
q dr
+ Re
|q| +
qr +
2
r
r
r2
r
|q|2
+ (R1 + R2 + R3 ).
:= −
2
First note
|q|2 q L 4/3 L 4/3
= q3L 4 L 4 .
x
t
t
x
Now consider q̃ R1 . Using the estimate
f (r ) L 2 ≤ Cr fr (r ) L 2
(Hardy’s inequality in R4 for radial functions), we have
≤ q L 4 L 4 R1 L 2 L 2 ≤ Cq3L 4 L 4 .
q̃ R1 L 4/3 ,L 4/3
x
t
t x
Next we note that |ν̄| = |J v Rv| = 1 − v32 = |Rv|. We consider next q̃ R3 . By
Hardy again,
∞
m
ν̄(1
−
mv
)
3
q̃
q̃ R3 L 4/3
≤
C
q
dr
4/3
2
x
r
r
L
ν̄ ν̄ 1 − v32 1/2
2
2
q
≤
C
q
≤
C
q
≤
C
4
(3.16)
L
L4
r 2
r 4
r 2 2 q L 4
L
L
L
1 + v3 1/2
2
−1/2
≤ C
+ q L 4 )q2L 4 ,
r 2 2 q L 4 ≤ C(s
L
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
483
where we used (3.14) again. Thus
(3.17)
−1/2
2
(s
≤
C
+
q
q̃ R3 L 4/3 ,L 4/3
4 )q 4 4/3
L
Lx L
x
x
t
−1/2 1/4
t2
≤C s
σ + q L 4t L 4x q L 4 L 4 .
t
x
It remains to estimate q̃ R2 . We rewrite R2 , using integration by parts,
∞
m
m
m
ν̄r − 2 ν̄ q dr
R2 = − Re ν̄q − Re
r
r
r
r
∞
mv3
m(1 + mv3 )
m
ν̄
q dr,
q̄ +
= − Re ν̄q + Re
r
r
r2
r
where we used ν̄r = −v3 q̄ − mr v3 ν̄. So
∞ 2 ν̄ |q|
q
q̃ R2 L 4/3 ,L 4/3
q̄
dr
≤
C
q
+
4/3 4/3
r 4/3 4/3 x
t
r
r
L t ,L x
L t ,L x
∞
ν̄
+
q dr q
4/3 4/3 .
r2
r
Lt
,L x
The last term is estimated as follows:
∞
ν̄ ν̄
q
q q L 4 q dr ≤ C
4/3
x
2
4/3 4/3
r
r L 2x
r
L t ,L x
L
t
ν̄ 2 ≤ C
r 4 q L 4x 43
Lx
Lt
(3.18)
−1/2
≤ C (s
+ q L 4x )q2L 4 L 4/3
x
t
≤ C s−1/2 σ 1/4 + q L 4t L 4x q2L 4 L 4 ,
t
x
where we used the same computations as in (3.16) and (3.17). The first term can be
treated in a similar manner, leading to the same estimate as in (3.18). The estimate
for the second term has been done already.
By returning now to (3.13) and using the above estimates, we establish (3.9).
Next we need to estimate the derivative of q̃ in order to establish (3.10). Denote
w := ∂xi q̃ for i = 1, 2. Then w solves
iwt + w =
(3.19)
m(1 + v3 )(mv3 − m − 2)
w
2
r
m(1 + v3 )(mv3 − m − 2)
+
q̃
r2
xi
m(v3 )r
m(v3 )r
w+
q̃ + N (q)w + N (q)xi q̃.
+
r
r
xi
484
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Using the previous estimates, we can estimate the various terms involving w in the
right-hand side:
m(1 + v3 )(mv3 − m − 2) −1
2
w
• 4/3 4/3 ≤ C(s + q L 4x )w L 4x L 4/3
t
r2
Lt
Lx
≤ C s−1 σ 1/2 + q2L 4 L 4 ∇ q̃ L 4t L 4x
t
x
m(v3 )r −1
• ≤ Cs−1 σ 1/2 ∇ q̃ L 4t L 4x
r w 4/3 4/3 ≤ Cs w L 4x L 4/3
t
Lt Lx
• N (q)w L 4/3 L 4/3
≤ C(s−1/2 σ 1/4 + q L 4t L 4x )q L 4t L 4x ∇ q̃ L 4t L 4x
x
t
Now the other terms. First note that |∇ q̃|2 ∼ |q̃|2 + |q̃/r |2 , and thus
q̃ r p 2 ≤ C∇ q̃ L p (R2 )
L (R )
for any 1 ≤ p ≤ ∞. Due to (3.14) and (3.15), we have
(1 + v3 )(mv3 − m − 2)
q̃ 4/3 4/3
2
r
x
Lt Lx
i
(v3 )r (1 + v3 ) ≤C r 2 q̃ 4/3 4/3 + r 3 q̃ 4/3 4/3
Lt Lx
L L
t x
q̃ 1 + v3 (v3 )r ≤ C
r 2 + r 2 2 r 4 4/3
Lx
≤ C s −1 + q2 4 ∇ q̃ L 4
Lx
x
Lx
Lx
Lt
4/3
Lt
≤ C s−1 σ 1/2 + q2L 4 L 4 ∇ q̃ L 4t L 4x .
t
x
Next we consider
m(v3 )r
(v3 )r (v3 )rr q̃
q̃
≤
C
q̃
+
4/3 4/3 .
4/3 4/3
r 2 4/3 4/3 r
r
xi
Lt Lx
Lt Lx
Lt Lx
The first term in the right side can be estimated as before:
(v3 )r −1 1/2
r 2 q̃ 4/3 4/3 ≤ Cs σ ∇ q̃ L 4t L 4x .
Lt
Lx
Recalling (h 1 )r = −(m/r )h 1 h 3 , (h 3 )r = (m/r )h 21 , and h 1 /r, h 3 bounded, we find
2
2 1 h1h3 z2
γ h 1 r
h1
|(v3 )rr | ≤ 2 m
+ (h 1 z 2,r )r − m
+ (γr h 3 )r + m
s
r r
r
r r
s
r
2 r
C h 1 h 1 z 2 h 1 (z 2 )r 2
+ |h 1 zrr | + |zzrr | + |zr |
.
+
2 +
≤ 2 s
r r
r
r
s
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
485
We estimate term by term:
2 1
q
h1
r
• s2
4/3
r r s
r L 4/3
t Lx
2
2
1 h 1 h 1 |h 3 | r
q
≤ C
≤ Cs−1 σ 1/2 ∇ q̃ L 4t L 4x ,
s2 r 2 + r 2 s
4/3
r L 4/3
L
t x
1 h 1 |z 2 | h 1 |(z 2 )r | r
q
• s
s2 r 2 +
4/3
r
r L 4/3
L
t x
−1 z q ≤ C
+
z
≤ Cs−1 σ 1/2 ∇ q̃ L 4t L 4x ,
s
2 r
L
x
r L 2x
r L 4x L 4/3
t
1
q
r
2
• s 2 |zr | s
4/3
r L 4/3
t Lx
1
2 q 2
−1 1/2
≤ C
∇ q̃ L 4t L 4x .
s zr L 4x r L 4 4/3 ≤ C q L 4t L 4x + s σ
x L
t
For the remaining term, using (3.7),
1
q
r
• 2 [(|h 1 | + |z|)zrr ]
4/3
s
s
r L 4/3
L
t x 1
r q
≤ C
z
2 (|h 1 | + |z|)
rr
L
s
x
s r L 4x L 4/3
t q r q
≤ C
qr L 2x + r 2 (|h 1 | + |z|) s r 4 4/3
Lx
L L
x t
r
q
2
−1
2 + C
q L 4x + s q L 2x (|h 1 | + |z|) s r 4 ≤ C q2 8/3
Lt
L 8x
+ s−3/2 σ 3/4 ∇ q̃ L ∞
2
t Lx
Lx
4/3
Lt
+ C q2L 4 L 4 + s−1 σ 1/2 ∇ q̃ L 4t L 4x ,
t
x
where we used, first by (3.6) with p = 8,
(|h 1 | + |z|) r q ≤ C (|h 1 | + |z|)(r/s) q L 8
8
x
s r L 4x
r
Lx
≤ Cq2L 8 + Cs −3/2 1 + q2L 2 ,
x
and then (|h 1 | + |z|)(r/s)(q/r ) L 4x ≤ Cq/r L 4x .
x
486
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
It remains to estimate N (q)xi q:
ν̄ q 2
+
|q|
q
≤
C
|
+
N (q)xi q̃ L 4/3 L 4/3
|q
r
r r 4/3 4/3
x
t
L L
t x q ν̄ 2
≤ C
|q| + r q 2 |qr | + r 4 4/3
Lx
Lx Lt
2 2
ν ≤ C
q L 4x + r 4 ∇ q̃ L 4x 4/3
Lt
Lx
≤ C s−1 σ 1/2 + q2L 4 L 4 ∇ q̃ L 4t L 4x .
t
x
Now applying Strichartz estimates to (3.19) and using the estimates established
above, we obtain
∇ q̃ L ∞,2 ∩L 4 L 4 ≤ ∇ q̃(τ ) L 2x + C (s−1 + s−2 )σ 1/2 + q2X (Q) ∇ q̃ L ∞,2 ∩L 4 L 4 .
t,x
t
t,x
x
t
x
This completes the proof of Lemma 3.1.
4 Technical Lemmas
In this section we collect some of the technical lemmas used in the proof of the
main theorem in the previous sections.
4.1 Some Inequalities for Radial Functions
We begin with some inequalities for radial functions.
L EMMA 4.1
(i) Let 0 ≤ σ < 1, and suppose f ∈ H 1 (R2 ) is radial. Then
∞
1−σ ∞
σ
∞ 2
f
2
2
(4.1)
r dr ≤ Cσ
f r dr
| fr | r dr .
r 2σ
0
0
0
We have limσ →1− Cσ = ∞ and the estimate is false if σ = 1.
(ii) Let 0 ≤ σ < 1, and suppose f ∈ H 1 (R2 ). Then
1−σ σ
f2
2
2
(4.2)
d x ≤ Cσ
| f | dx
|∇ f | d x .
|x|2σ
R2
R2
(iii) Suppose f ∈
(4.3)
1
Hloc
(R2 )
f L ∞ (R2 )
R2
is radial with fr , f /r ∈ L 2 (R2 ). Then
∞ f2
2
| fr | + 2 r dr .
≤C
r
0
P ROOF : We first show that
∞ 2
1/2 ∞
1/2
∞ 2
f (r )
f (r )
1
2
(4.4)
r dr ≤
r dr
( fr (r )) r dr
.
r 2σ
1 − σ 0 r 4σ −2
0
0
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
487
Indeed, by changing the order of integration, we get
∞ 2
∞
∞
f (r )
r
r dr = −
[ f 2 (s)]s ds dr
r 2σ
r 2σ r
0
0
s
∞
r
f (s) f s (s)ds
dr
= −2
2σ
0
0 r
∞
1
=−
s 2−2σ f (s) f s (s)ds
1−σ 0
∞
1/2 ∞
1/2
1
s 2−4σ f 2 (s)s ds
( f s (s))2 s ds
.
≤
1−σ 0
0
In particular, if σ = 12 , (4.4) immediately implies (4.1). We also note that estimate
(4.1) is immediate in the case σ = 0.
Let σi = 1 − (1/2i ) where i ≥ 0 is an integer. From estimate (4.4) with
σ = σi+1 , we have
∞ 2
f (r )
r dr
r 2σi+1
0
∞ 2
1/2 ∞
1/2
f (r )
i+1
2
r dr
( fr (r )) r dr
, i = 0, 1, 2, . . . .
≤2
r 2σi
0
0
Iterating this estimate, we obtain
∞ 2
f (r )
r dr
(4.5)
r 2σn+1
0
∞
1/2n+1 ∞
1−1/2n+1
2
2
≤ Cn+1
f (r )r dr
( fr (r )) r dr
0
0
√
for some constant Cn+1 . (One can solve Cn+1 = 2
Cn and C0 = 1 to get
−n
Cn+1 = 22n+2 , which is certainly not the best constant.)
It remains to consider the general case σ ∈ [0, 1). Let k ≥ 0 be an integer with
σk ≤ σ < σk+1 . There exists 0 ≤ θ < 1 such that σ = θσk+1 + (1 − θ)σk =
1 − (2 − θ)/2k+1 . Using the Hölder inequality and (4.5), we get
∞
θ ∞ 2
1−θ
∞ 2
f
f2
f
r dr ≤
r dr
r dr
r 2σ
r 2σk+1
r 2σk
0
0
0
∞
1−σ ∞
σ
2
2
≤ Cσ
f (r )r dr
( f ) (r )r dr ,
n+1
0
θ
Ck+1
Ck1−θ .
0
This completes the proof of the first estimate (4.1).
where Cσ =
To see that this estimate fails at the endpoint σ = 1, fix a smooth, nonnegative,
nondecreasing function η(r ), supported in ( 12 , ∞), and with 1 − η(r ) supported in
[0, 23 ). Then it is easy to check that f δ (r ) := η(r/δ) − η(r ) provides a counterexample to the endpoint estimate as δ → 0. Note that f δ (0) = 0 for all δ.
488
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
The second estimate (4.2) is an immediate consequence of (4.1). Using polar
coordinates, we obtain
2π ∞
| f (r, θ)|2
r dr dθ
r 2σ
0
0
1−σ ∞
σ
2π ∞
2
2
| f (r, θ)| r dr
|∂r f (r, θ)| r dr dθ.
≤ Cσ
0
0
0
Using |∂r f | ≤ |∇ f | , we obtain estimate (4.2) from Hölder’s inequality.
For the third estimate (4.3), we introduce the new variable y defined by r = e y
and denote g(y) = f (e y ) = f (r ). Then it is immediate that
∞
∞
f 2 (r )
2
(|g (y)|2 + |g(y)|2 )dy.
| fr (r )| +
r dr =
2
r
0
−∞
By Sobolev embedding, we have g L ∞ (R) ≤ Cg H 1 (R) . Transforming back to
the original variable completes our proof.
2
2
L EMMA 4.2 Let g : R2 → C be radial and bounded with g, g ∈ L loc , 2 < p < ∞.
Assume (∂r − m/r )g(r ) ∈ L p (R2 ) for some m ≥ 1. Then g(r )/r ∈ L p (R2 ) and
g(r ) m
≤
C
−
g(r
)
∂
r
r p 2
p 2.
r
p
L (R )
L (R )
P ROOF : Let 0 < r1 < r2 < ∞ and denote A = {x ∈ R2 : r1 < |x| < r2 }.
Consider
r2 m g p−2 ḡ
gr − g I := −2π Re
r dr.
r
r
r
r1
p−1
On one hand, I ≤ Cg/r L p (A) gr − (m/r )g L p (A) by the Hölder inequality. On
the other hand,
p
p
g
2π |g(r )| p r2
p − 2
g
I = m −
−
p−2
r L p (A)
p r
p
r L p (A)
r1
p
2 2π |g(r2 )| p
g
≥ m−1+
−
.
p r L p (A)
p r2p−2
Thus
p−1 p
2π |g(r2 )| p m 2 g
g
≤
+ .
m−1+
gr − g p r L p (A)
p r2p−2
r L p (A) r L p (A)
This gives a bound for g(r )/r L p (A) uniformly in r1 and r2 . Hence g(r )/r ∈
L p (R2 ). As r2 → ∞ and r1 → 0, we get
p−1 p
g
2 m
g
gr − g m−1+
≤
,
p r L p (R2 )
r L p (R2 )
r L p (R2 )
where we used p > 2 and the boundedness of g. This completes the proof.
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
489
Remark 4.3. It is essential to assume g is bounded, as can be seen by the example
g(r ) = r m . If we assume in the above lemma that g(r ) = o(1) as r → ∞, then we
also have
g(r ) 1
m
r 2 2 ≤ m ∂r − r g(r ) 2 2 .
L (R )
L (R )
p
Using Lemma 4.2, we prove an L -version of Lemma 2.5.
L EMMA 4.4 Let 2 < p < ∞. There exists > 0 such that if f (r ) is a radial
function satisfying | f, h 1 X | ≤ f X , then
f
fr L p (R2 ) + r p 2 ≤ C(L 0 f L p (R2 ) + L 0 f L 2 (R2 ) ).
L (R )
Recall L 0 f := fr + (m/r )h 3 f .
P ROOF : We note first that it suffices to prove
f
≤ C(L 0 f L p + L 0 f L 2 ),
r p
L
since
fr L p
m
f
m
≤ C L 0 f L p + ≤ C fr + h 3 f + h 3 f r
r
r
Lp
Lp
p .
L
Let ϕ : [0, ∞) → R be a standard cutoff function with
0
0 ≤ ϕ ≤ 1,
ϕ(r ) ≡ 1 for r < 1,
ϕ(r ) ≡ 0 for r > 2,
p
p
∞
∞
f ϕ
f (1 − ϕ) p
r dr +
r dr ≡ C(I + II).
f r dr ≤ C
r r r
∞
0
0
We consider the second term II. Since 1 − ϕ ≡ 0 if r < 1, we have
∞ 2
∞
f (1 − ϕ) p
f
p−2
(1 − ϕ)r dr
r dr ≤ f L ∞ (R2 )
II =
r r
0
0
2
f
p−2
p−2
2
≤ C f L ∞ (R2 ) r 2 2 ≤ C f L ∞ (R2 ) L 0 f L 2 (R2 ) ,
L (R )
where we used Lemma 2.5.
Next we consider the term I. Using Lemma 4.2, we have
p
f ϕ p
≤ C ( f ϕ)r − m f ϕ r p
p
r
L
L
p p
m
mh
3
p
≤ C ϕr f L p + fr + r f ϕ p + r (1 + h 3 ) f ϕ p .
L
L
490
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Since ϕr is supported only on (1, 2), and (1 + h 3 )/r is bounded, we have
f ϕ p
p
p
r p ≤ C( f L ∞ + L 0 f L p ).
L
Since, by Lemma 2.5, f L ∞ ≤ CL 0 f L 2 , we obtain
p
f
≤ C(L 0 f p p + L 0 f p 2 ),
L
r p
L
L
completing the proof.
4.2 Some Harmonic Map Estimates
Here we prove some facts about the family Om of m-equivariant harmonic
maps.
The first lemma shows that if m-equivariant harmonic maps h = h 0,1 and h α,s
are close in the sense of energy, then α and s are also close to 0 and 1, respectively.
L EMMA 4.5 Let 0 < s < ∞ and −π ≤ α < π. There exists > 0 and C > 0
such that if R2 |∇(emθ R (h −h s,α ))|2 d x < δ 2 for any δ < , then |α|+|s −1| ≤ Cδ.
P ROOF : Consider the case s > 1 (the case s < 1 can be treated in the same
way). We first note that


h 1 (r ) − h 1 (r/s) cos α
h − h s,α =  −h 1 (r/s) sin α  .
h 3 (r ) − h 3 (r/s)
Our assumption is
|∇(emθ R (h − h s,α ))|2 d x
R2
= 2π
0
∞
m2
s,α 2
s,α 2
|∂r (h − h )| + 2 |R(h − h )| r dr < δ 2 .
r
For 0 ≤ r ≤ 1, we have (s 2m − 1)(1 − r 2m ) ≥ 0, which, when rearranged, yields
r
2s m
h1
≤ 2m
h 1 (r ) ≤ h 1 (r ), 0 ≤ r ≤ 1.
s
s +1
Using this inequality, we find
2
mθ R
s,α
2
δ ≥ |∇(e (h − h ))| d x ≥ 2π
2
r dr
∂r h 3 (r ) − h 3 r
s
1
0
R2
1
= 2π
0
2
m2 2
1 2 r
h 1 (r ) − 2 h 1
r dr
r2
s
s
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
491
2
m2 2
2 r
(r
)
−
h
r dr
h
1
1
2
s
0 r
(s 2m − 1)4 1 h 41 (r )
≥ 2πm 2 2m
r dr
(s + 1)4 0 r 2
(s 2m − 1)4
= C 2m
.
(s + 1)4
≥ 2π
1
It follows that |s − 1| ≤ Cδ 1/2 if δ is sufficiently small. Now g(s) := ∂r (h 3 (r ) −
h 3 (r/s))2L 2 is a smooth function of s with g(1) = g (1) = 0 and g (1) > 0,
and so by Taylor’s theorem, we have g(s) ≥ C(s − 1)2 for some C > 0, and for
|s − 1| ≤ Cδ 1/2 , δ sufficiently small. Thus |s − 1| ≤ Cδ, as required.
Next consider the second component of h − h s,α :
2
∞
r 2
2
2
δ > 2π sin (α)
∂r h 1 s r dr ≥ C sin (α)
0
and so |sin(α)| < Cδ for sufficiently small δ.
Finally, use
r
(1 − cos(α))h 1 (r ) = (h − h s,α )1 + cos(α) h 1
− h 1 (r )
s
together with the previous results to arrive at 1 − cos(α) ≤ Cδ, from which (for
α ∈ [−π, π)) |α| ≤ Cδ follows.
The next lemma is a bound on the curvature of the family Om of m-equivariant
harmonic maps.
L EMMA 4.6 There are ε > 0 and C > 0 such that if
emθ R (h s1 ,α1 (r ) − h s2 ,α2 (r )) Ḣ 1 < ,
then setting s̄ := 12 [s1 + s2 ], ᾱ := 12 [α1 + α2 ], and h̄ := 12 [h s1 ,α1 + h s2 ,α2 ], we have
(4.6)
emθ R (h̄(r ) − h s̄,ᾱ (r )) Ḣ 1 < Cemθ R (h s1 ,α1 (r ) − h s2 ,α2 (r ))2Ḣ 1 .
P ROOF : By rotating and rescaling, we may assume (s1 , α1 ) = (1, 0). If is
sufficiently small, Lemma 4.5 gives (taking α2 ∈ [−π, π))
(4.7)
|s2 − 1| + |α2 | ≤ Cemθ R (h 1,0 (r ) − h s2 ,α2 (r )) Ḣ 1 ≤ C.
Now set s(t) := s̄ + (t/2)(s2 − 1), α(t) := ᾱ + (t/2)α2 , and φ(t) := h s(t),α(t) .
Then
1
h̄ − h s̄,ᾱ = [φ(−1) − φ(0) + φ(1) − φ(0)]
2
1
0
1
φ (t)dt −
φ (t)dt
=
2 0
−1
1 1 1 1 t =
[φ (t) − φ (−t)]dt =
φ (τ )dτ.
2 0
2 0 −t
492
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Using (4.7), we have
emθ R φ (τ ) Ḣ 1 ≤ C[(s2 − 1)2 + α22 ] ≤ Cemθ R (h 1,0 (r ) − h s2 ,α2 (r ))2Ḣ 1
and (4.6) follows.
Our next lemma gives L ∞ -smallness for Ḣ 1 -small perturbations of harmonic
maps.
L EMMA 4.7 For u ∈ m , set s = s(u) and α = α(u), and write
r
r
+ξ
.
u(r, θ) = emθ R v(r ),
v(r ) = eα R h
s
s
There exists > 0 and C > 0 such that if δ0 < and
2
m
2
mθ R s,α 2
h Ḣ 1 < δ02 ,
(4.8)
ξr L 2 (R2 ) + Rξ 2 2 = u − e
r
L (R )
then
ξ L ∞ ≤ Cδ0 .
P ROOF : Without loss of generality, we may assume s = 1 and α = 0. It
follows immediately from (4.8) and Lemma 4.1 that
ξi ∞ ≤ Cδ0 ,
i = 1, 2.
For ξ3 , we have, as yet, only (ξ3 )r L 2 < δ0 , and so our aim is to show that
ξ3 L ∞ ≤ Cδ0 . Under our change of variable ξ̃ (y) := ξ(m log(r )), it suffices to
prove that ξ̃3 L 2 (R) ≤ Cδ0 , since ξ̃3 L ∞ (R) ≤ Cξ̃3 H 1 (R) . By the continuity
and boundary conditions of v(r ) (for u ∈ m ), there must exist y0 ∈ R such that
ṽ3 (y0 ) = 0. Note that since h̃ 3 (y) = tanh(y), we have
2 2
2
2
2 ṽ j − h̃ j ≤ C(|ξ1 | + |ξ2 |),
(4.9)
|ṽ3 (y) − tanh (y)| = j=1
and in particular,
tanh2 (y0 ) ≤ C(ξ1 L ∞ + ξ2 L ∞ ) < Cδ0 .
So for −1 ≤ y ≤ 1,
(4.10)
y
|ṽ3 (y) − tanh(y)| = (ṽ3 − tanh (y))dy − tanh(y0 )
y0
≤ C(ṽ3 − tanh L 2 + δ0 ) ≤ Cδ0 ,
1/2
1/2
and in particular, for δ0 sufficiently small, |ṽ3 (±1)| > (1/2) tanh(1). Then with
the aid of Lemma 2.3, for δ0 sufficiently small, we have |ṽ3 (y)| > (1/4) tanh(1)
for |y| ≥ 1. Estimate (4.9) then yields
(ṽ3 (y) − tanh(y))2 dy ≤ Cδ02 ,
|y|≥1
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
493
which also gives us sup|y|≥1 |ṽ3 (y) − tanh(y)| ≤ Cδ0 (Sobolev embedding), and in
particular |ṽ3 (1) − tanh(1)| ≤ Cδ0 . Finally, we get the same result for |y| < 1 by
integrating the derivative:
1
|ṽ3 (y) − tanh(y)| ≤ (ṽ3 − tanh )dy + Cδ0 < Cδ0 .
y
This completes the proof.
4.3 Perturbation Is Bounded by q
Here we prove estimates used in Section 3. We show that z(r ) is controlled by
q(r ), where, recall,
m
vr − J v Rv = q1 ê + q2 J v ê, Drv ê ≡ 0,
r
and
r
r
αR
+ξ
, ξ = z 1 e + z 2 J h e + γ h.
h
(4.11)
v(r ) = e
s
s
L EMMA 4.8 Let 2 ≤ p < ∞. For δ sufficiently small,
z
1−2/ p
(4.12)
q L p (R2 ) + q L 2 (R2 ) ),
zr L p (R2 ) + r p 2 ≤ C(s
L (R )
q 2
(4.13) zrr L 2 (R2 ) ≤ C sqr L 2 (R2 ) + s + sq L 4 (R2 ) + q L 2 (R2 ) ,
r 2 2
L (R )
and
q 1
1
2
≤C
+ q L 4 (R2 ) + q L 2 (R2 ) .
+ qr L 2 (R2 ) + s
r L 2 (R2 )
s
(4.14) u Ḣ 2 (R2 )
P ROOF : We will first show the following:
z
(4.15)
zr L 2 (R2 ) + r 2 2 ≤ Cq L 2 (R2 ) .
L (R )
By (4.11), we have
2m
m
h 3 γ h + ξ3 ξ.
r
r
Since z X ≤ CL 0 z L 2 by (2.25), it suffices to prove that L 0 z ≤ Cq L 2 . We
first show that ξr L 2 + ξ/r L 2 ≤ Cz X . Indeed, since (J h e)r = −(m/r )h 1 h
and h r = (m/r )h 1 J h e, we find
m
m
ξr = zr e − z 2 h 1 h + γr h + γ h 1 J h e.
r
r
2
Therefore, since γ = O(|z| ) and γr = O(|z||zr |), we obtain
z
z
ξr L 2 ≤ zr L 2 + + z L ∞ zr L 2 + z L ∞ r 2,
r L2
L
(4.16)
e−α R s(q ê)(sr ) = (L 0 z)e + (γ h)r +
494
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
where we used the boundedness of h. By (2.23), we have z L ∞ ≤ Cδ, which can
be chosen sufficiently small to yield ξr L 2 ≤ Cz X . In a similar manner, we can
show ξ/r L 2 ≤ Cz/r L 2 ≤ Cz X . Combining, we obtain
ξ (4.17)
ξr L 2 + r 2 ≤ Cz X .
L
Now we are ready to prove L 0 z L 2 ≤ Cq L 2 . Using again γ = O(|z|2 ),
γr = O(|z||zr |), and the boundedness of h, we find
z
ξ 2m
m
(γ h)r +
+ Cξ L ∞ h 3 γ h + ξ3 ξ ≤ Cz L ∞ zr L 2 + r 2
r
r
r L2
L2
L
ξ ≤ Cξ L ∞ z X + L 2
r
≤ Cξ L ∞ z X ≤ Cξ L ∞ L 0 z L 2 ,
where we used (4.17) and z L ∞ ≤ ξ L ∞ . Thus we have
2m
m
+
γ
h
+
ξ
L 0 z L 2 ≤ sq(s·) L 2 + h
ξ
(γ
h)
r
3
3 r
r
L2
≤ q L 2 + Cξ L ∞ L 0 z L 2 .
Since ξ L ∞ ≤ Cξ X ≤ Cz X ≤ Cδ can be taken sufficiently small, the above
inequality implies
(4.18)
L 0 z L 2 ≤ Cq L 2 ,
which completes the proof of (4.15). Similarly, for any p with 2 ≤ p < ∞, we
have
z
p
p
∞
p
L 0 z L ≤ C sq(s·) L + z L zr L + r p
L
1−2/ p
≤C s
q L p + z L ∞ (L 0 z L p + L 0 z L 2 ,
where we used Lemma 4.4. Since z L ∞ can be taken sufficiently small, by using (4.18), we finally have L 0 z L p ≤ C(s 1−2/ p q L p + q L 2 ), completing the
proof of (4.12).
Next we prove (4.13). We first show
L(ze) (4.19)
zrr L 2 ≤ C ∂r L(ze) L 2 + + L 0 z L 2 .
r L 2
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
495
Indeed, recalling L(ze) = zr e + (m/r )h 3 ze, we have
1 m
1
m
zr e + h 3 zr e
∂r + + h 3 zr e = ∂r +
r
r
r
r
1
m
m
= ∂r +
L(ze) − h 3 ze + h 3 zr e
r
r
r
2
2
m
1
m
L(ze) − 2 h 21 ze + 2 z 2 h 3 h 1 h.
= ∂r +
r
r
r
Set
η := emθ R ze
and
H := emθ R h.
Since ∂ j η = D jH η − (∂ j H · η)H and (h 3 )r = (m/r )h 21 , we have
e
−mθ R
D jH D jH η
m2
1
zr e − 2 h 23 ze
=
∂r +
r
r
1
m
m2
(zr e) + z 2 h 1 h − 2 h 23 ze
= ∂r +
r
r
r
1 m
m
m
= ∂r + + h 3 (zr e) − h 3 L(ze) + z 2 h 1 h,
r
r
r
r
and therefore we obtain
2
H H D D η 2 = ∂r + 1 zr e − m h 2 ze
j
j
3 L
2
r
r
L2
m
1
=
∂r + r L(ze) − r h 3 L(ze)
m2 2
m2
m − 2 h 1 ze + 2 z 2 h 3 h 1 h + z 2 h 1 h 2
r
r
r
L
L(ze) z
1
≤ C ∂r +
+ z L 2 + L(ze) + r 2
r
r
2
2
L
L
L L(ze) ≤ C ∂r L(ze) L 2 + r 2 + L 0 z L 2 ,
L
where we used |h 1 /r | bounded. Since ∂ j η = D jH η − (∂ j H · η)H , we have
η = ∂ j ∂ j η = ∂ j (D jH η − (∂ j H · η)H )
= ∂ j D jH η − ∂ j ((∂ j H · η)H )
= D jH D jH η − (∂ j H · D jH η)H − (∂ j ∂ j H · η)H − (∂ j H · ∂ j η)H
− (∂ j H · η)∂ j H.
496
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
Therefore, we obtain
η L 2 ≤ D jH D jH η L 2 + (∂ j H · D jH η)H L 2 + (∂ j H · ∂ j η)H L 2
+ (∂ j ∂ j H · η)H L 2 + (∂ j H · η)∂ j H L 2
η
H H
≤ D j D j η L 2 + C ∇η L 2 + r 2
L
z
≤ D jH D jH η L 2 + C zr L 2 + r 2 .
L
Thus
(4.20)
η Ḣ 2 ≤ Cη L 2
L(ze) ≤ C ∂r L(ze) L 2 + r + L 0 z L 2 .
L2
Let
A = (ze)rr −
(ze)r
m 2 ze
+ 2 ,
r
r
B = (ze)rr ,
E=
m R(ze)r
m Rze
.
−
r
r2
Then direct calculations show
2
xy
y
(emθ R ze)x x = emθ R − 2 A + B − 2 2 E ,
r
r
2
x
xy
mθ R
mθ R
(e
− 2 A+ B+2 2 E ,
ze) yy = e
r
r
2
x
y2
mθ R
mθ R x y
(e
ze)x y = e
A+
− 2 E .
r2
r2
r
Each of A, B, and E can be expressed in terms of combinations of second derivatives of η = emθ R ze. This implies, in particular, estimate (4.19).
It remains to control ∂r L(ze) L 2 + L(ze)/r L 2 . Noting that êr = −(vr · ê)v
and (J v ê)r = (vr · ê)v, so
(q ê)r = qr ê − q1 (vr · ê)v + q2 (vr · ê)v,
and recalling vr (r ) = eα R (1/s)(h r (r/s) + ξr (r/s)), we have
q 1
· ∂r (q ê) L 2 ≤ C qr L 2 + + q zr
r L2
s
s L 2
q 2
−1
2
≤ C qr L 2 + + q L 4 + s q L 2 ,
r L2
SCHRÖDINGER FLOW NEAR HARMONIC MAPS
497
where we used (4.12). Taking the derivative of Equation (4.16), we get
q 2
2
∂r L(ze) L 2 ≤ C sqr L 2 + s r 2 + sq L 4 + q L 2
L
m
m
.
+
(γ
h)
+
2
γ
h
+
ξ
h
ξ
rr
3
3
r
r
r
r L2
We consider first (γ h)rr . Using |h r | + |r h rr | < C, we have
(γ h)rr L 2 ≤ C|γrr | + |γr h r | + |γ h rr | L 2
z 2
≤ C z L ∞ zrr L 2 + zr L 4 + |zr | + r L 2
≤ C(z L ∞ zrr L 2 + sq2L 4 + q2L 2 + q L 2 ).
Next we consider ((m/r )h 3 γ h)r . In a similar manner, we find
2
m
z
γ γr z
h
γ
h
≤
C
≤
C
+
z
+
4 3
r
L
r
r 4
r 2 r 2
r 4
r L2
L
L
L
2
2
≤ C sq L 4 + q L 2 .
For the term (ξ3 ξ/r )r , we have the estimate
m
2
2
r ξ3 ξ 2 ≤ C sq L 4 + q L 2 .
r L
Following a similar procedure for L(ze)/r L 2 , we obtain
L(ze) ∂r L(ze) L 2 + r L 2
q 2
2
≤ C sqr L 2 + s + z L ∞ zrr L 2 + sq L 4 + q L 2
r 2
L
q L(ze) ≤ C sqr L 2 + s r 2 + z L ∞ ∂r L(ze) L 2 + r 2
L
L
+ sq2L 4 + q2L 2 ,
using (4.19). Since z L ∞ can be taken sufficiently small, we conclude
L(ze) q 2
≤ C sqr L 2 + s + sq L 4 + q L 2 ,
(4.21) ∂r L(ze) L 2 + r L 2
r L2
having used the smallness of q L 2 . Combining (4.21) with (4.19) completes the
proof of (4.13).
498
S. GUSTAFSON, K. KANG, AND T.-P. TSAI
It remains to prove (4.14). Since u(x) = e(mθ+α)R (h(r/s) + ξ(r/s)), it is
straightforward to check that for δ sufficiently small,
C
u Ḣ 2 ≤ (1 + η Ḣ 2 )
s
and so (4.14) follows from (4.20) and (4.21). The proof of Lemma 4.8 is complete.
Acknowledgments. We thank Nai-Heng Chang, Jalal Shatah, and Chongchun
Zeng for sharing with us their results on self-similar singular solutions, which are
not used in this paper but inspired us nonetheless. The first and third authors are
partially supported by grants from the Natural Sciences and Engineering Research
Council of Canada. The second author was partially supported by a Pacific Institute
for the Mathematical Sciences Postdoctoral Fellowship.
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S TEPHEN G USTAFSON
University of British Columbia
Department of Mathematics
Vancouver, BC V6T 1Z2
CANADA
E-mail: gustaf@math.ubc.ca
TAI -P ENG T SAI
University of British Columbia
Department of Mathematics
Vancouver, B.C. V6T 1Z2
CANADA
E-mail: ttsai@math.ubc.ca
Received April 2005.
K YUNGKEUN K ANG
Sungkyunkwan University
Department of Mathematics
and Institute of Basic Science
Suwon 440-746
SOUTH KOREA
E-mail: kkang@skku.edu