Global well-posedness for the Schr\"odinger map problem with small Besov norm

0.17 MB·

by  Benjamin Dodson

Global well-posedness for the Schr¨odinger map
7 problem with small Besov norm
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2
Benjamin Dodson

n
a
January 31, 2017
J

7
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Abstract: In this paper we prove a global result for the Schr¨odinger map

]
problem with initial data with small Besov norm at critical regularity.
P
A
. 1 Introduction
h
t
a
In this paper we consider the Schr¨odinger map initial value problem
m
[ ∂ φ=φ×∆φ, on R×Rd,
t
(1.1)
1 φ(0)=φ ,
0
v
5 in dimensions d ≥ 3, where φ : Rd×R → S2 ֒→ R3 is a continuous function.
9 TheSchr¨odingermapequationarisesinferromagnetismastheHeisenbergmodel
1
for the ferromagnetic spin system.
8
0
. In this paper we are concernedwith the issue of globalwell- posedness of (1.1)
1
for data which is small in a critical space under the scaling.
0
7
1 Observe that for any λ>0, if φ(t,x) solves (1.1), then
:
v
i φ(λ2t,λx) (1.2)
X
r also solves (1.1). Now for Q∈S2, Z+ ={0,1,2,...},and σ ∈[0,∞), define the
a
space
Hσ ={f :Rd →R3, |f(x)|≡1 a.e. f −Q∈Hσ}, (1.3)
Q
where Hσ is the usual inhomogeneous Sobolev space. This metric has the in-
duced distance
dσQ(f,g)=kf −gkHσ(Rd). (1.4)
Also define the metric spaces
H∞ =∩σ∈Z+Hσ, and HQ∞ =∩σ∈Z+HQσ. (1.5)
1
Thescalinglaw(1.2)preservestheH˙d/2andH˙d/2homogeneousSobolevnorms,
Q
where
kfk =kF(f)(ξ)·|ξ|σk , (1.6)
H˙σ L2
kfk =kf −Qk , (1.7)
H˙σ H˙σ
Q
and F :L2 →L2 is the usual Fourier transform
F(f)(ξ)=c e−ix·ξf(x)dx. (1.8)
dZ
Rd
(1.1) is globally well - posed for data which is small in H˙d/2.
Theorem 1.1 (Global regularity) Assume d ≥ 2 and Q ∈ S2. Then there
exists ǫ (d) > 0 such that for any φ ∈ H∞ with kφ −Qk ≤ ǫ (d), then
0 0 Q 0 H˙d/2 0
there is a unique solution
φ=S (φ )∈C(R:H∞) (1.9)
Q 0 Q
of the initial value problem (1.1). Moreover,
supkφ(t)−Qk ≤Ckφ −Qk , (1.10)
H˙d/2 0 H˙d/2
t∈R
and for any T ∈[0,∞) and σ ∈Z ,
+
sup kφ(t)kHσ ≤C(σ,T,kφkHσ). (1.11)
Q Q
t∈[−T,T]
This theorem was proved in dimensions d ≥ 4 in [2], and then for dimensions
d≥2 in [3]. [3] also proved a uniform global bound.
Theorem 1.2 (Uniform bounds and well - posedness) Assumed≥2,Q∈
S2, and σ ≥ d. Then there exists 0 < ǫ (d,σ ) ≤ ǫ (d) such that if φ ∈ H∞
1 2 0 1 0 Q
with kφ−Qk ≤ ǫ (d,σ ), then the global solution φ(t) to (1.1) with initial
H˙d/2 0 1
data φ satisfies
0
d
supkφ(t)−Qk ≤C kφ −Qk , ≤σ ≤σ . (1.12)
Hσ σ 0 Hσ 1
t∈R 2
In addition, the solution operator admits a continuous extension from
Bσ ={φ∈H˙d/2−1∩H˙σ :kφ−Qk ≤ǫ } (1.13)
ǫ0 Q H˙d/2 0
to C(R;H˙σ∩H˙d/2−1).
Q
2
The case when d=2 is particularly interesting, due to the fact that a solution
to (1.1) conserves the quantities
E (t)= |φ(t)−Q|2dx, (1.14)
0 Z
Rd
and
d
E (t)= |∂ φ(t,x)|2dx. (1.15)
1 Z m
Rd X
m=1
However,ingeneraltheproofsintheorems1.1and1.2aremoredifficultinlower
dimensions. Indeed, [3] utilized the caloric gauge of [18] to analyze dimension
d=2, because the Coulomb gauge used in [3] was not strong enough.
In this paper we will extend theorems 1.1 and 1.2 to data which is small in
Besov - type norms.
Theorem 1.3 Suppose φ ∈H˙d/2 and suppose that ψ(x)∈C∞(Rd) is a radial
0 0
function supported on 1 ≤ |x| ≤ 4 and φ(x) = 1 on 1 ≤ |x| ≤ 2. Furthermore,
4 2
suppose that for some ǫ (d,kφ k ),
0 0 H˙d/2
supkψ(2jξ)|ξ|d/2·F(φ−Q)(ξ)kL2(Rd) ≤ǫ0(d,kφ0kH˙d/2). (1.16)
j
Then the results of theorems 1.1 and 1.2 hold.
In this paper we follow [1], [2], [3], [10], [11], [12], and [14] and prove a priori
bounds on the derivative of the Schr¨odinger map rather than the Schr¨odinger
mapitself. We usethe Coulombgauge,asis typicallyusedinanalysisofhigher
dimensions, see for example [2].
The main new ingredient to this paper is the bilinear estimates obtained from
the interaction Morawetz estimates. In [7], applying the bilinear estimates of
[13]to the variousSchr¨odingermapgaugesgavebilinearestimates for solutions
to (1.1) in dimensions d ≥ 2. These results were used in [8] to improve the
results of [15], [16], and [17].
Hereweareabletousebilinearargumentsinordertoprovetheorem1.3. These
arguments greatly simplify the function spaces that are needed in the analysis.
2 Bilinear Virial estimates
The work in this paper will heavily rely on bilinear estimates for solutions to
the linear Schr¨odinger equation
iu +∆u=0. (2.1)
t
3
Theorem 2.1 (Interaction Morawetz estimate) If u solves (2.1) then
k|∇|3−2d|u|2k2L2t,x(R×Rd) .kuk2L∞t L2x(R×Rd)kuk2L∞t H˙x1/2(R×Rd). (2.2)
Proof: This theorem was proved in [6] when d=3. [19] subsequently extended
the result to dimensions d ≥ 4. [13] and [5] independently proved theorem 2.1
in dimensions d=1,2. (cid:3)
Ofcourse,[6],[19],[13],and[5]alsoprovedtheorem2.1fornonlinearSchr¨odinger
equationsaswell. [13]alsomadesomeveryimportantinsightsintothebehavior
of linear solutions as well. In particular, [13] observed that one may compute
the interaction Morawetz estimate for two different solutions to (2.1), yielding
a bilinear estimate.
Theorem 2.2 (Bilinear virial estimate) Suppose u and v are solutions to
(2.1). Then
|∂ (u(t,x ,...,x )v(t,x ,y ,...,y ))|2dxdydt
Z Z 1 1 d 1 2 d
x1=y1
.kuk2 kvk2 +kvk2 kuk2 .
L∞t H˙1/2(R×Rd) L∞t L2x(R×Rd) L∞t H˙1/2(R×Rd) L∞t L2x(R×Rd)
(2.3)
In particular, suppose that u and v are solutions to (2.1) and P is the usual
j
Littlewood - Paley projection operator. That is, let ψ(x) = 1 for |x| ≤ 1 and
ψ(x) = 0 for |x| > 2 be a smooth, radially symmetric function, decreasing as
r →∞. Then let
ψ (x)=ψ(2−jx)−ψ(2−j+1x), (2.4)
j
and let P be the Fourier multiplier given by
j
F(P f)(ξ)=[ψ(2−jξ)−ψ(2−j+1ξ)]F(f)(ξ). (2.5)
j
Then by elementary Fourier analysis,
(P f)(x)=2jd ψˇ(2j(x−y))f(y)dy, (2.6)
j Z
whereψˇis asmoothfunction thatisrapidly decreasingfor|x| large,thatis, for
any integer N,
|ψˇ(x)|. (1+|x|)−N. (2.7)
N
Since the Littlewood - Paley operator is a Fourier multiplier, if u solves (2.1)
then P u also solves (2.1), so
j
4
|∂ (P u(t,x ,x ,...,x )P v(t,x ,y ,...,y ))|2dxdydt
Z Z 1 j 1 2 d k 1 2 d
x1=y1
.kP uk2 kP vk2 +kP vk2 kP uk2 .
j L∞t H˙1/2(R×Rd) k L∞t L2x(R×Rd) k L∞t H˙1/2(R×Rd) j L∞t L2x(R×Rd)
(2.8)
But then by (2.6), (2.7), (2.8), and Ho¨lder’s inequality, if k ≤j,
k∂ (P uP v)k2 .2(d−1)k2jkP uk2 kP vk2 .
1 j k L2 (R×Rd) j L∞L2(R×Rd) k L∞L2(R×Rd)
t,x t x t x
(2.9)
Now since there is absolutely nothing special about the direction e ∈ Sd−1,
1
averagingover all directions ω ∈S1 gives
k∇(P uP v)k2 .2(d−1)k2jkP uk2 kP vk2 .
j k L2 (R×Rd) j L∞L2(R×Rd) k L∞L2(R×Rd)
t,x t x t x
(2.10)
Thus, if j ≥ k+10, then by the Fourier support of P u·P v, and Bernstein’s
j k
inequality,
k(P uP v)k2 .2(d−1)k2−jkP uk2 kP vk2 .
j k L2 (R×Rd) j L∞L2(R×Rd) k L∞L2(R×Rd)
t,x t x t x
(2.11)
Also, by theorem 1.1, when d≥3, and j−10≤k ≤j,
k(P uP v)k2 .2(d−1)k2−jkP uk2 kP vk2 .
j k L2 (R×Rd) j L∞L2(R×Rd) k L∞L2(R×Rd)
t,x t x t x
(2.12)
Remark: This will not workin dimensions d=1,2 because we can merely say
thatP u·P v is supportedon|ξ|.2k whenj−10≤k ≤j, sowhen the power
j k
of |∇|3−2d is positive, then (2.12) does not follow directly from theorem 1.1 and
the aboveanalysis. It is for this technicalreasonthat this paper only addresses
dimensions d≥3.
Itisappropriatetopointoutthat(2.11)hasbeenprovento be truein[4]using
Fourier analysis. In fact, [4] and subsequent Fourier analytic proofs in higher
dimensions all predate the bilinear virial arguments of [13]. However, in this
case, it is quite useful to examine the bilinear estimates throughthe lens of the
virial identity. In [13] the interaction Morawetz for theorem 2.2 is
(x−y)
M(t)= |v(t,y)|2 1 Im[u¯∂ u](t,x)dxdy
Z |(x−y) | 1
1
(2.13)
(x−y)
+ |u(t,y)|2 1 Im[v¯∂ v](t,x)dxdy.
Z |(x−y) | 1
1
Averaging over all directions in Sd−1, suppose u is the solution to
5
iu +∆u=N , (2.14)
t 1
and v is the solution to
iv +∆v =N . (2.15)
t 2
For any Rd, for some c , for any y ∈Rd,
d
x·ω x·y
(y·ω)dω =c . (2.16)
ZSd−1 |x·ω| d |x|
Without loss of generality suppose x=|x|e . Then,
1
1 x·ω 1 x·(−ω) x ω2
y·ω+ (y·(−ω))= 1 1 y , (2.17)
2|x·ω| 2|x·(−ω)| |x ||ω | 1
1 1
whichimplies(2.16). Thenbytheorem2.2andaveragingoverω ∈Sd−1,taking
∂ f =ω·∇f, and x =x·ω,
ω ω
|∂ (u(t,y)v(t,x))|2dxdydtdω
Z Z ω
Sd−1 xω=yω
.kuk2 kvk2 +kvk2 kuk2
L∞t H˙1/2(R×Rd) L∞t L2x(R×Rd) L∞t H˙1/2(R×Rd) L∞t L2x(R×Rd)
(x−y)
+ |v(t,y)|2 ·Im[u∇N ](t,x)dxdydt
Z |(x−y)| 1
(x−y)
+ |v(t,y)|2 ·Im[N ∇u](t,x)dxdydt
Z |(x−y)| 1
(x−y)
+ |u(t,y)|2 ·Im[v∇N ](t,x)dxdydt
Z |(x−y)| 2
(x−y)
+ |u(t,y)|2 ·Im[N ∇v](t,x)dxdydt
Z |(x−y)| 2
(x−y)
+ Re[vN ](t,y) ·Im[u¯∇u](t,x)dxdydt
Z 2 |(x−y)|
(x−y)
+ Re[uN ](t,y) ·Im[v¯∇v](t,x)dxdydt.
Z 1 |(x−y)|
(2.18)
Then by Ho¨lder’s inequality and averaging over ω ∈Sd−1, for any R>0,
6
1
|∇(u(t,x)v(t,x+h))|2dxdhdt
RZ Z Z
h∈Rd:|h|≤R Rd
.kuk2 kvk2 +kvk2 kuk2
L∞t H˙1/2(R×Rd) L∞t L2x(R×Rd) L∞t H˙1/2(R×Rd) L∞t L2x(R×Rd)
(x−y)
+ |v(t,y)|2 ·Im[u∇N ](t,x)dxdydt
Z |(x−y)| 1
(x−y)
+ |v(t,y)|2 ·Im[N ∇u](t,x)dxdydt
Z |(x−y)| 1
(x−y)
+ |u(t,y)|2 ·Im[v∇N ](t,x)dxdydt
Z |(x−y)| 2
(x−y)
+ |u(t,y)|2 ·Im[N ∇v](t,x)dxdydt
Z |(x−y)| 2
(x−y)
+ Re[vN ](t,y) ·Im[u¯∇u](t,x)dxdydt
Z 2 |(x−y)|
(x−y)
+ Re[uN ](t,y) ·Im[v¯∇v](t,x)dxdydt.
Z 1 |(x−y)|
(2.19)
Therefore, for k ≤j,
k(P uP v)k2 .2(d−1)k2−jkP uk2 kP vk2
j k L2 (R×Rd) j L∞L2(R×Rd) k L∞L2(R×Rd)
t,x t x t x
(x−y)
+2(d−1)k2−2j |P v(t,y)|2 ·Re[P u∇P N ](t,x)dxdydt
Z k |(x−y)| j j 1
(x−y)
+2(d−1)k2−2j |P v(t,y)|2 ·Re[P N ∇P u](t,x)dxdydt
Z k |(x−y)| j 1 j
(x−y)
+2(d−1)k2−2j |P u(t,y)|2 ·Re[P v∇P N ](t,x)dxdydt
Z j |(x−y)| k k 2
(x−y)
+2(d−1)k2−2j |P u(t,y)|2 ·Re[P N ∇P v](t,x)dxdydt
Z j |(x−y)| k 2 k
(x−y)
+2(d−1)k2−2j Re[P vP N ](t,y) ·Re[P u¯∇P u](t,x)dxdydt
Z k k 2 |(x−y)| j j
(x−y)
+2(d−1)k2−2j Re[P uP N ](t,y) ·Re[P v¯∇P v](t,x)dxdydt.
Z j j 1 |(x−y)| k k
(2.20)
Thisisausefulestimatesinceitopensupthepossibilityofintegratingbyparts,
andmovingaderivativetoamoreadvantageousposition. Thisopportunitywill
proveto be quite useful, since we will follow the analysis in [2]. That is, if φ be
a solution to (1.1), let
ψ =∇φ=(ψ ,...,ψ )=(∂ φ,...,∂ φ). (2.21)
x 1 d 1 d
Then, choosing to work in the Coulomb gauge, for m=1,...,d,
7
d d d
(i∂ +∆ )ψ =−2i A ·∂ψ +(A + A2)ψ −i Im(ψ ψ¯)ψ , (2.22)
t x m l l m 0 l m m l l
X X X
l=1 l=1 l=1
where
d
∂
A =− kIm(ψ ψ¯ ), (2.23)
l l k
∆
X
k=1
and
d d
∂∂ 1
A = l m(Re(ψ¯ψ ))− ( ψ ψ¯ ). (2.24)
0 l m m m
∆ 2
X X
l,m=1 m=1
3 Growth of Besov norms
Now we utilize frequency envelopes to control the growth of the Besov norm.
The frequency envelopes were introduced in [18] for the study of wave maps.
Frequency envelopes majorize the size of a Littlewood - Paley projection of a
function,whileatthesametimesmoothingoutthedifferencesinsizeatdifferent
frequency levels. Let δ >0 be a small constant, say δ = 1, and let
4
αj(0)=sup2−δ|j−k|2k(d−22)kPkψx(0)kL2(Rd), (3.1)
k
αj =sup2−δ|j−k|2k(d−22)kPkψxkL∞L2(R×Rd), (3.2)
k t x
βj =sup2−(δ/2)|j−k|2k(d−42)kPkψxkL4 (R×Rd), (3.3)
k t,x
and
γj =sup2−δ|j−k|2k(d−22)(sup221(k−l)k(Pkψx)(Plψx)kL2 (R×Rd))
k l≤k t,x
+sup2−δ|j−k|2k(d−22)(supsup R12k2(d−2)lZ |Pkψx(t,x)|2|Plψx(t,x+h)|2dxdydt)1/2.
k l≤kR>0 h∈Rd,|h|≤R
(3.4)
Remark: From the definition,
β2 .γ . (3.5)
j j
Also observe that for any j,
kPjψx(0)kH˙ d−22 ≤αj(0), (3.6)
and by Young’s inequality,
8
kψ (0)k2 ∼ α (0)2. (3.7)
x H˙ d−22 X j
j
Now if hf,gi is the inner product
Re f(x)g(x)dx, (3.8)
Z
then
kP ψ k2 =hP ψ ,P ψ i, (3.9)
j x L2(Rd) j x j x
x
so by (2.22),
d
d
hP ψ ,P ψ i=−4hP ( A∂ ψ ),P ψ i
dt j x j x j l l x j x
X
l=1
(3.10)
d d
−2hiP ((A + A2)ψ ),P ψ i−2hP ( Im(ψ ψ¯)ψ ),P ψ i.
j 0 l x j x j x l l j x
X X
l=1 l=1
Now by Fourier support arguments,
d
sup2−2δ|j−k|2k(d−2) |hP ( Im(ψ ψ¯)ψ ),P ψ i|dt
Z k x l l k x
k R X
l=1
.sup2−2δ|j−k|2k(d−2)kP ψ k kP ψ k3
k k x L4t,x ≥k−10 x L4t,x
+sup2−2δ|j−k|2k(d−2)k(P ψ )(P ψ )k k(P ψ )(P ψ )k
k x ≤k−10 x L2 k−10≤·≤k+10 x ≤k−10 x L2
k t,x t,x
.β4+γ2 .γ2.
j j j
(3.11)
By a similar calculation,
d
1
sup2−2δ|j−k|2k(d−2) |hiP (− ( ψ ψ¯ )ψ ),P ψ i|dt.β4+γ2 .γ2.
k ZR k 2 X m m x k x j j j
m=1
(3.12)
Now split
∂ ∂ ∂ ∂
l mRe(ψ¯ψ )= l mRe((P ψ¯)(P ψ ))
l m ≥k−20 l ≥k−20 m
∆ ∆
∂∂ ∂ ∂
+2 l mRe((P ψ¯)(P ψ ))+ l mRe((P ψ¯)(P ψ )).
≥k−20 l ≤k−20 m ≤k−20 l ≤k−20 m
∆ ∆
(3.13)
Because ∂l∂m is a bounded Fourier multiplier,
∆
9
∂ ∂
sup2−2δ|j−k|2k(d−2) |h l m(Re(P ψ¯)(P ψ ))ψ ,P ψ i|dt
Z ∆ ≥k−20 l ≥k−20 m x k x
k
.sup2−2δ|j−k|2k(d−2)kP ψ k2 kP ψ k kP ψ k
k ≥k−20 x L4t,x k x L4t,x ≥k−10 x L4t,x
+sup2−2δ|j−k|2k(d−2)kP ψ k2 k(P ψ )(P ψ )k .β4+γ2.
k ≥k−20 x L4t,x k x ≤k−10 x L2t,x j j
(3.14)
Meanwhile, because (P ψ¯)(P ψ ) is supported on |ξ| ≤ 2k−15, by
≤k−20 l ≤k−20 m
(2.7) we compute
sup2−2δ|j−k|2(d−2)k 2ld|ψˇ(2l(x−y))||P ψ (t,y)|2
Z Z l≤·≤k−20 x
k X Rd
l≤k−15
×|P ψ (t,x)||P ψ (t,x)|dxdydt (3.15)
k x k−10≤·≤k+10 x
.γ2 2(d−1)l 2−(d−2)m2−(d−2)k2−k .γ2.
j j
X X
l≤k−20 l≤m≤k−20
Also by (2.7),
sup2−2δ|j−k|2(d−2)k 2ld|ψˇ(2l(x−y))||P ψ (t,y)|2
Z Z ≤l x
k X Rd
l≤k−15
×|P ψ (t,x)||P ψ (t,x)|dxdydt
k x k−10≤·≤k+10 x
.sup2−2δ|j−k|2(d−2)k k(P ψ )(P ψ )k k(P ψ )(P ψ )k .γ2.
k x ≤l x L2 k−10≤·≤k+10 x ≤l x L2 j
k X t,x t,x
l≤k−15
(3.16)
Therefore, combining (3.12) - (3.16) with an interpolation argument proves
sup2−2δ|j−k|2k(d−2)|hiP ((A )ψ ),P ψ i|dt.γ2. (3.17)
k 0 x k x j
k
Next, integrating by parts, since d ∂A =0
l=1 l l
P
d
−4h A ·∂(P ψ ),P ψ i=0. (3.18)
l l k x k x
X
l=1
To estimate
d
−4h [P ,A ]∂ ψ ,P ψ i, (3.19)
k l l x k x
X
l=1
observethatifmisaFouriermultipliersatisfying|∇m(ξ)|. 1 and|m(ξ)|.1,
|ξ|
then by the fundamental theorem of calculus, if |η|<<|ξ|,
|ξ||m(ξ+η)fˆ(η)gˆ(ξ)−m(ξ)fˆ(η)gˆ(ξ)|.|η||fˆ(η)||gˆ(ξ)|. (3.20)
10