The focusing cubic NLS on exterior domains in three dimensions

0.33 MB·English

by  Rowan Killip

THE FOCUSING CUBIC NLS ON EXTERIOR DOMAINS IN
THREE DIMENSIONS
5
1
ROWANKILLIP,MONICAVISAN,ANDXIAOYIZHANG
0
2

n
a
J Abstract. WeconsiderthefocusingcubicNLSintheexteriorΩofasmooth,

1 compact, strictly convex obstacle in three dimensions. We prove that the
2 threshold for global existence and scattering is the same as for the prob-
lem posed on Euclidean space. Specifically, we prove that if E(u0)M(u0) <

] E(Q)M(Q) and k∇u0k2ku0k2 < k∇Qk2kQk2, the corresponding solution to
P the initial-value problem with Dirichlet boundary conditions exists globally
A andscatterstolinearevolutions asymptoticallyinthefutureandinthepast.
Here,Q(x)denotes thegroundstateforthefocusingcubicNLSinR3.
.
h
t
a 1. Introduction
m
We considerthe focusingcubic nonlinearSchr¨odingerequationinthe exteriorof
[
a strictly convex obstacle Ωc ⊂R3 with Dirichlet boundary conditions

1
v iu +∆ u=−|u|2u
t Ω
2 (1.1) (NLSΩ)
6 (u(0,x)=u0(x).
0 Hereu:R×Ω→Cand−∆ denotestheDirichletLaplacian,whichisaself-adjoint
5 Ω
operator on L2(Ω) with form domain H1(Ω). We take initial data u ∈H1(Ω).
0 0 0 0
. When posed on the whole Euclidean space R3, the problem is scale invariant.
1
More precisely, the mapping
0
5 (1.2) u(t,x)7→uµ(t,x):=µu(µ2t,µx) with µ>0
1
v: leaves the class of solutions to NLSR3 invariant. This scaling also identifies the
i spaceH˙x1/2 asthecriticalspace. Since thepresenceofthe obstacledoesnotchange
X
the intrinsic dimensionality of the problem, we may regard equation (1.1) is being
r subcritical for data in H1(Ω).
a 0
Throughout this paper we restrict ourselves to the following notion of solution:
Definition 1.1 (Solution). Let I be a time interval containing the origin. A
function u : I ×Ω → C is called a (strong) solution to (1.1) if it lies in the class
C (I ,H1(Ω))∩L5 (I ×Ω) for any compact interval I ⊂ I, and it satisfies the
t ′ 0 t,x ′ ′
Duhamel formula
t
(1.3) u(t)=eit∆Ωu0+i ei(t−s)∆Ω(|u|2u)(s)ds
Z0
forallt∈I. TheintervalI issaidtobemaximalifthesolutioncannotbeextended
beyond I. We say u is a global solution if I =R.
In this formulation, the Dirichlet boundary condition is enforced through the
appearance of the linear propagator associated to the Dirichlet Laplacian.
1
2 R.KILLIP,M.VISAN,ANDX.ZHANG
Localwell-posednessforthedefocusingversionof (1.1)wasestablishedinseveral
works;see, for example,[1, 2, 7, 11]. Standardarguments relyingon the Strichartz
estimates proved in [7] and the equivalence of Sobolev spaces proved in [9] (see
Theorem 2.4) can be used to construct a local theory for (1.1).
Theorem 1.2 (Local well-posedness). Let u ∈H1(Ω). Then there exists a max-
0 0
imal time interval I such that (1.1) admits a strong solution on I. The interval I
and the solution u are uniquely determined by u . Moreover, the following hold:
0
• I is an open interval containing the origin.
• (Conservation laws) Mass and energy are conserved by the flow: for any t∈I,
M(u(t)):= |u(t,x)|2dx=M(u ),
0
ZΩ
E(u(t)):= 1|∇u(t,x)|2− 1|u(t,x)|4dx=E(u ).
2 4 0
ZΩ
• (Blowup) If supI <∞, then limsup ku(t)k =∞.
t supI H1(Ω)
• (Scattering) Suppose [0,∞) ⊆ I and→kuk 0 < ∞. Then u scatters
L5 ([0, ) Ω)
t,x ∞ ×
forward in time, that is, there exists u ∈H1(Ω) such that
+ 0
ku(t)−eit∆Ωu+kH1(Ω) →0 as t→∞.
0
• (Small data GWP) There exists η > 0 such that if ku k ≤ η , then the
0 0 H1(Ω) 0
0
solution u to (1.1) is global and satisfies
kukL5 (R Ω) .ku0kH1(Ω).
t,x × 0
In particular, the solution scatters in both time directions.
In this paper, we consider the global existence and scattering question for large
initialdata. Toputtheproblemincontext,letusfirstrecallsomeearlierresultsfor
the equivalent problem posed in the whole Euclidean space R3. In [5] it is shown
thatthefocusingcubicNLSonR3 isgloballywell-posedandscatterswheneverthe
initial data lies below the ground state threshold. To state this result explicitly,
let Q denote the unique, positive, spherically-symmetric, decaying solution to the
elliptic problem
∆Q−Q+Q3 =0.
The main result in [5] states that whenever
(1.4) E(u )M(u )<E(Q)M(Q) and k∇u k ku k <k∇Qk kQk ,
0 0 0 L2 0 L2 L2 L2
x x x x
the solution to NLSR3 is global and it satisfies global L5t,x spacetime bounds. For
spherically-symmetric initial data, this result was proved in the earlier work [6].
Note that in the Euclidean case, the quantity E(u)M(u) is scale invariant and
conserved in time. Throughout this paper, we will refer to this quantity as the
H1/2-energy. Note that both of the assumptions on the initial data in (1.4) are
x
scale invariant.
By the variational characterization of Q, we know that the only functions f ∈
H1(R3) that satisfy
E(f)M(f)=E(Q)M(Q) and k∇fkL2(R3)kfkL2(R3) =k∇QkL2(R3)kQkL2(R3)
THE FOCUSING CUBIC NLS ON EXTERIOR DOMAINS IN THREE DIMENSIONS 3
areofthe formf(x)=eiθρQ(ρ[x−x ]) where θ ∈[0,2π), ρ∈(0,∞), andx ∈R3.
0 0
In the presence of an obstacle, there are no functions f ∈H1(Ω) such that
0
E(f)M(f)=E(Q)M(Q) and k∇fkL2(Ω)kfkL2(Ω) =k∇QkL2(R3)kQkL2(R3).
Thiscanbeseeneasilybyextendingf tobeidenticallyequaltozeroontheobstacle.
The variational characterization of Q on R3 then yields that f must be Q up to
the symmetries of the equation; these functions, however, do not obey Dirichlet
boundary conditions.
We contend that even though (1.1) does not admit a direct analogue of the
soliton Q, the threshold for global well-posedness and scattering is still the same
as for the problem posed on R3. The main result of this paper verifies the positive
part of this claim:
Theorem 1.3 (Global well-posedness and scattering). Let u ∈H1(Ω) satisfy
0 0
(1.5) E(u )M(u )<E(Q)M(Q),
0 0
(1.6) k∇u0kL2(Ω)ku0kL2(Ω) <k∇QkL2(R3)kQkL2(R3).
Then there exists a unique global solution u to (1.1) and
(1.7) kukL5 (R Ω) ≤C M(u0),E(u0) .
t,x ×
In particular, u scatters in both time direc(cid:0)tions. (cid:1)
Observe that our bound in (1.7) depends jointly on the mass and energy of the
initial data, not simply their product. A bound depending only on M(u )E(u )
0 0
could be obtained (with significantadditional complexity) by incorporatingrescal-
ingintoourargumentsinthe styleof[10]; however,weknowofnoapplicationthat
benefits from this stronger assertion.
Due to the convexity of the curve ME = E(Q)M(Q), we may exhaust the
regionofthe mass/energyplanewhere(1.5)holdsby sub-levelsets ofthe following
one-parameter family of free energies: for each 0<λ<∞, we define
Fλ(u ):=E(u )+λM(u ).
0 0 0
To be precise, setting
Fλ :=2 λM(Q)E(Q)
∗
one easily sees that p
(1.8) {u :E(u )M(u )<E(Q)M(Q)}= {u :Fλ(u )<Fλ}
0 0 0 0 0
∗
0<λ<
[∞
Note that Fλ = Fλ(Qµ) where µ = λM(Q)/E(Q) and Qµ is the rescaling of
Qdefinedvia(1∗.2). Correspondingly,wenotethatFλ isthefreeenergyofasoliton
p
solution to the cubic NLS in R3. ∗
In view of (1.8), we see that Theorem 1.3 follows from the following result.
Theorem 1.4. Let u ∈H1(Ω) satisfy (1.6). If
0 0
Fλ(u )=E(u )+λM(u )<Fλ
0 0 0
∗
for some 0<λ<∞, then there exists a unique global solution u to (1.1) and
kukL5 (R Ω) ≤C(λ,Fλ(u0)).
t,x ×
4 R.KILLIP,M.VISAN,ANDX.ZHANG
As alluded above, we believe that (1.5) and (1.6) represent the sharp threshold
in our setting, just as they do for the problem posed in R3. More precisely, in [6]
it is shown that radial initial data u ∈H1(R3) obeying (1.5) and
0
k∇u0kL2(R3)ku0kL2(R3) >k∇QkL2(R3)kQkL2(R3)
lead to solutions that blow up in finite time. This is proved via the usual concave
virialargument,whichdoesnotadaptdirectlytooursetting—theboundaryterm
doesnothaveafavorablesign. Itisnaturaltoimaginethatitshouldbepossibleto
embed Euclidean blowup solutions into the exterior domain case via perturbative
arguments; however, our current understanding of the structure of this blowup is
notquite sufficienttopushthis through. Nevertheless,wecanshowthatourresult
is sharp in terms of uniform spacetime bounds:
Proposition 1.5. Fix λ ∈ (0,∞). There exists a sequence of global solutions
u ∈C H1(R×Ω) that satisfy (1.6) and
n t 0
Fλ(un)րFλ and kunkL5 (R Ω) →∞ as n→∞.
∗ t,x ×
This is provedin Section7 by choosinga sequence u that closely models Qµ in
n
shape, but is centered far from the obstacle.
Acknowledgements. R. K. was supported by NSF grant DMS-1265868. M. V.
was supported by the Sloan Foundation and NSF grants DMS-0901166 and DMS-
1161396. X. Z. was supported by the Simons Foundation.
2. Preliminaries
2.1. Notation and useful lemmas. We write X .Y or Y &X to indicate X ≤
CY for some constant C >0. We use the notation X ∼Y whenever X .Y .X.
Throughout this paper, Ω denotes the exterior domain of a compact, smooth,
strictly convex obstacle in R3. Without loss of generality, we assume that 0 ∈ Ωc
and Ωc ⊂B(0,1). For any x∈R3 we use d(x):=dist(x,Ωc).
Withx ∈R3,weuseτ todenotethetranslationoperatorτ f(x):=f(x−x ).
0 x0 x0 0
Throughout this paper, χ will denote a smooth cutoff in R3 satisfying
1, if |x|≤ 1
(2.1) χ(x)= 4
(0, if |x|> 1.
2
We will use the following refined version of Fatou’s lemma due to Brezis and
Lieb.
Lemma 2.1 (Refined Fatou, [3]). Let 0<p<∞ and assume that {f }⊆Lp(Rd)
n
with limsup kf k <∞. If f →f almost everywhere, then
n n p n
→∞
|f |p−|f −f|p−|f|p dx→0 as n→∞.
n n
ZRd(cid:12) (cid:12)
In particular, kfnkpp(cid:12)(cid:12)−kfn−fkpp →kfkpp. (cid:12)(cid:12)
(cid:12) (cid:12)
We will use the following heat kernel estimate due to Q. S. Zhang.
Lemma 2.2 (Heat kernel estimate, [12]). Let Ω denote the exterior of a smooth,
compact, convex obstacle in Rd for d≥3. Then there exists c>0 such that
|et∆Ω(x,y)|. √td(dxi)am ∧1 √td(dyi)am ∧1 e−c|x−ty|2t−d2,
∧ ∧
uniformly for x,y ∈Ω. If ei(cid:16)ther x∈/ Ω o(cid:17)r(cid:16)y ∈/ Ω, then e(cid:17)it∆Ω(x,y)=0.
THE FOCUSING CUBIC NLS ON EXTERIOR DOMAINS IN THREE DIMENSIONS 5
There is a naturalfamily ofSobolev spacesassociatedto powersofthe Dirichlet
Laplacian. Our notation for these is as follows:
Definition 2.3. For s ≥ 0 and 1 < p < ∞, let H˙s,p(Ω) and Hs,p(Ω) denote the
D D
completions of C (Ω) under the norms
c∞
kfkH˙Ds,p(Ω) :=k(−∆Ω)s/2fkLp and kfkHDs,p(Ω) :=k(1−∆Ω)s/2fkLp.
When p=2 we write H˙s(Ω) and Hs(Ω) for H˙s,2(Ω) and Hs,2(Ω), respectively.
D D D D
When Ω is replaced by R3, these definitions lead to the classical H˙s,p(R3) and
Hs,p(R3) families of spaces. Note that the classical Hs,p(Ω) spaces are defined as
0
subspacesofHs,p(Ω),whichareinturndefinedasquotientsofthespacesHs,p(R3).
ThusthespacesHs,p(Ω)havenodirectconnectiontothefunctionalcalculusofthe
0
Dirichlet Laplacian. It is a well-known (but nontrivial) theorem that Hs,p(Ω) =
0
Hs,p(Ω) for 0<s< 1 and again for 1 <s<1+ 1.
D p p p
The inter-relation of homogeneous Sobolev spaces on non-compact manifolds
throws up additional complications. The case s= 1 captures already the question
of boundedness of Riesz transforms, which is a topic of on-going investigation.
Motivated by applications to NLS, in [9] we investigated when the two notions of
Sobolev spaces are equivalent in the case of exterior domains. The advantage of
such an equivalence is two fold: Powers of the Dirichlet Laplacian commute with
the linear evolution eit∆Ω, while fractional product and chain rules (needed for
treating the nonlinearity) have already been proved for spaces defined via powers
of the Euclidean Laplacian. Our findings in [9] are summarized in the following
sharp result about the equivalence of Sobolev spaces.
Theorem 2.4 (Equivalence of Sobolev spaces, [9]). Let d ≥ 3 and let Ω be the
complement of a compact convex body Ωc ⊂Rd with smooth boundary. Let 1<p<
∞. If 0≤s<min{1+ 1,d} then
p p
(−∆Rd)s/2f Lp∼d,p,s (−∆Ω)s/2f Lp for all f ∈Cc∞(Ω).
Theorem(cid:13) 2.4allowsu(cid:13)stotransf(cid:13)erdirectlyse(cid:13)veralkeyresultsfromtheEuclidean
(cid:13) (cid:13) (cid:13) (cid:13)
setting to exterior domains. One example is the Lp-Leibnitz (or product) rule for
first derivatives:
Corollary 2.5 (Fractional product rule). For all f,g ∈C (Ω), we have
c∞
k(−∆Ω)12(fg)kLp(Ω) .k(−∆Ω)12fkLp1(Ω)kgkLp2(Ω)+kfkLq1(Ω)k(−∆Ω)21gkLq2(Ω)
with the exponents satisfying 1<p,p ,q <∞, 1<p ,q ≤∞, and 1 = 1 + 1 =
1 2 2 1 p p1 p2
1 + 1.
q1 q2
The following simple lemma will be frequently used in this paper.
Lemma 2.6. Let φ∈H1(R3), χ be as in (2.1), and R →∞. Then
n
χ Rxn φ H1(R3) .kφkH1(R3),
(cid:13)lim(cid:0) (cid:1)1−(cid:13) χ x φ =0.
(cid:13)n (cid:13) Rn H1(R3)
→∞
(cid:13)(cid:2) (cid:0) (cid:1)(cid:3) (cid:13)
(cid:13) (cid:13)
6 R.KILLIP,M.VISAN,ANDX.ZHANG
Proof. We only prove the second assertion. By Ho¨lder’s inequality,
1−χ x φ
Rn H1(R3)
(cid:13)(cid:13)(cid:2) . (cid:0)1−(cid:1)χ(cid:3) (cid:13)(cid:13)Rxn φ L2(R3)+ 1−χ Rxn ∇φ L2(R3)+Rn−1 φ(∇χ) Rxn L2(R3)
.(cid:13)(cid:13)kφ(cid:2)kL2(|x(cid:0)|&Rn(cid:1))(cid:3)+(cid:13)(cid:13)k∇φkL2(|(cid:13)(cid:13)x(cid:2)|&Rn)+(cid:0) k∇(cid:1)χ(cid:3)kL3(cid:13)(cid:13)(R3)kφkL6(|x|∼R(cid:13)(cid:13)n). (cid:0) (cid:1)(cid:13)(cid:13)
The claim now follows from the monotone convergence theorem. (cid:3)
By exploiting the functional calculus for self-adjoint operators, one can define
the Littlewood–Paley projections adapted to ∆ . Just like their Euclidean coun-
Ω
terparts, these operators obey Bernstein estimates.
Lemma 2.7 (Bernstein estimates). Let 1 < p < q ≤ ∞ and −∞< s < ∞. Then
for any f ∈C (Ω), we have
c∞
kPΩ fk +kPΩfk .kfk ,
N Lp(Ω) N Lp(Ω) Lp(Ω)
≤
kPΩNfkLq(Ω)+kPNΩfkLq(Ω) .N3(p1−q1)kfkLp(Ω),
≤
NskPNΩfkLp(Ω) ∼k(−∆Ω)s2PNΩfkLp(Ω).
2.2. Strichartz estimates, local smoothing, and the virial identity.
Strichartz estimates for domains exterior to a compact, smooth, strictly convex
obstaclewere provedby Ivanovici[7]; see also[2]. Ivanoviciobtainedthe full range
of Strichartz estimates known in the Euclidean setting, with the exception of the
endpoint L2L6. As we will only be using a finite collection of (non-endpoint)
t x
Strichartz norms in this paper, we can encapsulate everything into the following
Strichartz spaces: For ε>0 sufficiently small we define
6(2+ε)
S0(I):=L L2(I×Ω)∩L2+εL 2+3ε (I×Ω).
∞t x t x
Further, we define N0(I) as the corresponding dual Strichartz space and
N1(I):={f : f,(−∆Ω)12f ∈N0(I)}.
Additionally, in our discussion of solutions in the whole Euclidean space, it will be
convenient to use
S1(I):={u:I×R3 →C: (1−∆R3)12u∈L∞t L2x(I ×R3)∩L2tL6x(I ×R3)}.
Note that we will only use the S1(I) notation in the context of solutions in the
whole Euclidean space.
With these notations, the Strichartz estimates read as follows:
Theorem 2.8 (Strichartz estimates, [7]). Let I ⊂ R be a time interval and let
t ∈I. Then the solution u:I×Ω→C to
0
iu +∆ u=f
t Ω
satisfies
kuk ≤ku(t )k +kfk .
S0(I) 0 L2(Ω) N0(I)
In particular,
kuk .ku(t )k +kfk .
L5H1,1310(I Ω) 0 H01(Ω) N1(I)
t 0 ×
Using Theorems 2.8 and 2.4, and arguing in the usual manner (cf. [4]), one
obtains the following stability result for (1.1).
THE FOCUSING CUBIC NLS ON EXTERIOR DOMAINS IN THREE DIMENSIONS 7
Lemma2.9(Stability). Let Ωbetheexterior of acompact, smooth, strictlyconvex
obstacle in R3. Let I ⊂ R be a time interval and let u˜ be an approximate solution
to (1.1) on I×Ω in the sense that
i∂ u˜+∆ u˜=−|u˜|2u˜+e
t Ω
for some function e. Assume that
ku˜k ≤E and ku˜k ≤L
L∞H1(I Ω) L5 (I Ω)
t 0 × t,x ×
for some positive constants E and L. Let t ∈ I and u ∈ H1(Ω), and assume the
0 0 0
smallness conditions
ku˜(t )−u k ≤ε and kek ≤ε
0 0 H1(Ω) N1(I)
0
for some 0 < ε < ε = ε (E,L). Then there exists a unique strong solution u :
1 1
I×Ω→C to (1.1) with initial data u at time t=t satisfying
0 0
ku−u˜k ≤C(E,L)ε.
L5H1,3101(I Ω)
x 0 ×
We will also use the local smoothing estimate. The particular version we need
is [10, Lemma 2.13]:
Lemma 2.10 (Local smoothing). Let u=eit∆Ωu0. Then
|∇u(t,x)|2hR 1(x−z)i3dxdt.Rku k k∇u k
− 0 L2(Ω) 0 L2(Ω)
ZZR×Ω
uniformly for z ∈R3 and R>0.
A direct consequence of the local smoothing estimate is the following result,
which will be used in the proof of the Palais–Smale condition. For a similar state-
mentadaptedtotheenergy-criticalprobleminEuclideanspace,see[8,Lemma2.5].
Corollary 2.11. Given u ∈H1(Ω), we have
0 0
k∇eit∆Ωu0kL2t,x(|t−τ|≤T,|x−z|≤R) .R3109T110keit∆Ωu0kL315t,x(R×Ω)ku0kH3201(Ω).
Proof. We split the left-hand side according to low and high frequencies. To esti-
mate the low frequencies, we use Ho¨lder and Bernstein:
k∇eit∆ΩPΩNu0kL2 (t τ T,x z R) .T130R190Nkeit∆Ωu0kL5 (R Ω).
≤ t,x | − |≤ | − |≤ t,x ×
Toestimatethehighfrequencies,weusethelocalsmoothingestimateLemma2.10:
k∇eit∆ΩPΩNu0kL2 (t τ T,x z R) .R21N−12ku0kH1(Ω).
≥ t,x | − |≤ | − |≤ 0
The claim follows by summing these two estimates and optimizing N. (cid:3)
The last result in this subsection is a truncated virial inequality. Let φ be a
smooth radial cutoff function such that
|x|2, if |x|≤1
φ(x):=
(0, if |x|≥2.
For R≥1, let φ (x):=R2φ x .
R R
(cid:0) (cid:1)
8 R.KILLIP,M.VISAN,ANDX.ZHANG
Lemma 2.12 (Truncated Virial). Suppose 0∈Ωc and R>100diam(Ωc). Then
∂ Im u(t,x)∂ u(t,x)∂ φ (x)dx≥ 4|∇u(t,x)|2−3|u(t,x)|4dx
t k k R
ZΩ ZΩ
(2.2) −O R−2+ |u(t,x)|4+|∇u(t,x)|2dx .
(cid:16) Z|x|≥R (cid:17)
Proof. We will exploit the local momentum conservation identity
(2.3) ∂ Im(∂ uu¯)=−2∂ Re(∂ u∂ u¯)+ 1∂ ∆(|u|2)+ 1∂ (|u|4).
t k j k j 2 k 2 k
Integrating this against ∂ φ , we obtain
k R
∂ Im u¯∂ u∂ φ dx=−2Re ∂ (∂ u∂ u¯)∂ φ dx
t k k R j k j k R
ZΩ ZΩ
+ 1 ∂ ∆(|u|2)∂ φ dx+ 1 ∂ (|u|4)∂ φ dx.
2 k k R 2 k k R
ZΩ ZΩ
Wefirstseeklowerboundsoneachofthetermsappearingontheright-handside
of the equality above. From the divergence theorem and the fact that ∂ φ (x)=
jk R
2δ for |x|<R, we obtain
jk
−2Re ∂ (∂ u∂ u¯)∂ φ dx
j k j k R
ZΩ
=−2Re ∂ (∂ u∂ u¯∂ φ )dx+2Re ∂ u∂ u¯∂ φ dx
j k j k R k j jk R
ZΩ ZΩ
=2Re ∇u·∇φ ∇u¯·~ndσ(x)+2Re ∂ u∂ u¯∂ φ dx
R k j jk R
Z∂Ω ZΩ
≥2 |u |2(φ ) dσ(x)+4 |∇u|2dx−O |∇u|2dx .
n R n
Z∂Ω ZΩ (cid:16)Z|x|≥R (cid:17)
Here ~n denotes the outer normal to ∂Ω (i.e., ~n points into Ω), u := ∇u·~n, and
n
(φ ) :=∇φ ·~n. We have also used the fact that ∇u=u ~n, which follows from
R n R n
the Dirichlet boundary conditions.
Arguing similarly for the second term, we obtain
1 ∂ ∆(|u|2)∂ φ dx=−1 ∆(|u|2)(φ ) dσ(x)− 1 ∆(|u|2)∆φ dx
2 k k R 2 R n 2 R
ZΩ Z∂Ω ZΩ
=− |∇u|2(φ ) dσ(x)− 1 |u|2∆∆φ dx
R n 2 R
Z∂Ω ZΩ
≥− |∇u|2(φ ) dσ(x)−O(R 2).
R n −
Z∂Ω
The third term can be estimated as follows:
1 ∂ (|u|4)∂ φ dx=−1 |u|4∆φ dx≥−3 |u|4dx−O |u|4dx .
2 k k R 2 R
ZΩ ZΩ ZΩ (cid:16)Z|x|≥R (cid:17)
Putting all the pieces together and noting that ∇φ (x)=2x on ∂Ω, we deduce
R
∂ Im u¯∂ u∂ φ dx≥ 4|∇u|2−3|u|4dx+2 |∇u|2x·~ndx
t k k R
ZΩ ZΩ Z∂Ω
−O R−2+ |u|4+|∇u|2dx .
(cid:16) Z|x|≥R (cid:17)
Finally,as∂Ωisconvexwehavethatx·~n≥0,whichimmediatelyleadsto(2.2). (cid:3)
THE FOCUSING CUBIC NLS ON EXTERIOR DOMAINS IN THREE DIMENSIONS 9
2.3. Convergence results. The defects ofcompactnessinthe Strichartzinequal-
ity
ke−it∆ΩfkL5 (R Ω) .kfkH1(Ω)
t,x × 0
are the same as in the Euclidean case, namely, spacetime translations. (Scaling is
notanissuebecauseL5 hasdimensionalitystrictlybetweenthatofL2 andH˙1.) In
t,x x x
theEuclideancase,thesedefectsofcompactnessareassociatedtoexactsymmetries
of the equation. In our case, however, the obstacle breaks the space translation
symmetry. Correspondingly, our linear profile decomposition must handle possible
changesin geometry. This issue was systematicallystudied in [10] (where a scaling
symmetrywasalsopresent). Inthispaper,werecordonlytherelevantconvergence
results from [10], namely, when the obstacle is marching away to infinity relative
to the initial data. This scenario gives rise to the whole Euclidean space R3 as the
limiting geometry.
Proposition 2.13 (Convergence of domains, [10]). Suppose {x } ⊂ Ω are such
n
that |x |→∞ and write Ω :=Ω−{x }. For h∈C (R3) and Θ∈C (0,∞) we
n n n c∞ c∞
have
nl→im∞keit∆Ωnh−eit∆R3hkH˙−1(R3)∩H˙1(R3) =0,
nlim k[Θ(−∆Ωn)−Θ(−∆R3)]δ(y)kH˙−1(R3) =0,
→∞
1 1
nlim k(−∆Ωn)2h−(−∆R3)2hkL2(R3) =0,
→∞
lim keit∆Ωnh−eit∆R3hk 30 =0.
n→∞ L5t,x(R×R3)∩L5tLx11(R×R3)
The second limit above is uniform in y on compact subsets in R3.
Proof. The first two assertions follow from Proposition 3.6 in [10], which asserts
convergence in H˙ 1. This implies weak convergence in H˙1 which we can then
x− x
upgrade to strong convergence since
keit∆R3hkH˙1(R3) =khkH˙1(R3) =keit∆ΩnhkH˙1(R3)
by energy conservation for the free propagator.
The third relation is Lemma 3.7 from [10]. The last equation follows directly
fromTheorem4.1of[10],forexponentpair(5,30),andfrominterpolationbetween
11
it and Corollary 4.2 of that paper, for exponent pair (5,5). (cid:3)
Proposition 2.14 below is needed to prove asymptotic decoupling of parameters
in the linear profile decomposition. The two statements made by this proposition
are essentially Lemmas 5.4 and 5.5 in [10].
Proposition 2.14 (Weak Convergence, [10]). Assume Ω =Ω or Ω =Ω−{y }
n n n
with |y |→∞. Then the following two statements hold:
n
1. Let (t ,x )∈R×R3 satisfy |t |+|x |→∞ as n→∞. Then for f ∈C (Ω)
n n n n c∞
if Ω =Ω, or for f ∈C (R3) if Ω =Ω−{y },
n c∞ n n
τxneitn∆Ωnf ⇀0 weakly in H1(R3).
2. Let f ∈ H1(Ω ) be such that f ⇀ 0 weakly in H1(R3). Let t → t ∈ R.
n 0 n n n
Then ∞
eitn∆Ωnfn ⇀0 weakly in H1(R3).
10 R.KILLIP,M.VISAN,ANDX.ZHANG
The next lemma, the last for this subsection, will be used to prove decoupling
of the L4-norms in Proposition 3.1.
Lemma 2.15 (Weak dispersive estimate). Let φ ∈ C (Ω) and ψ ∈ C (R3). Let
c∞ c∞
Ω :=Ω−{y } with |y |→∞. Then for any sequence t →∞,
n n n n
lim keitn∆ΩφkL4 =0 and lim keitn∆ΩnψkL4 =0.
n x n x
→∞ →∞
Proof. To prove that the first limit is zero, we consider the function
F(t):= |eit∆Ωφ(x)|4dx.
ZΩ
From the Strichartz inequality, we know that F(t)∈L1(R); indeed,
t
keit∆ΩφkL4t,x(R×Ω) .kφkH˙01/4(Ω).
On the other hand, as
ddtF(t) . |eit∆Ωφ|3|eit∆Ω∆Ωφ|dx.k∆ΩφkL2xkeit∆Ωφk3L6x .kφk4H02(Ω),
Z
we se(cid:12)e that(cid:12)F is uniformly continuous. That the first limit is zero follows easily
(cid:12) (cid:12)
from these two facts.
For a Lipschitz function f : R →[0,∞) we have f(t)2 . f(s)ds. Combining
R
this with the argument above and the fourth part of Proposition 2.13, we derive
R
lim sup [eit∆Ωn −eit∆R3]ψ(x) 5dx=0.
n→∞ t ZR3
(cid:12) (cid:12)
Combining this with the L5/4 →(cid:12) L5 dispersive estim(cid:12)ate for the Euclidean propa-
x x
gator, we deduce that
|eitn∆Ωnψ(x)|5dx→0 as n→∞.
ZΩ
The claim now follows from the conservation of mass and Ho¨lder’s inequality. (cid:3)
2.4. Coercivity of the energy. The coercivity property, which is part of the
variationalcharacterizationofthegroundstate,playsanimportantrolethroughout
the proof. The version we use in this paper is a minor adaptation of the one in [6]
and is informed by our needs when proving Theorem 1.4.
Proposition 2.16. Fix λ>0 and let u ∈H1(Ω) satisfy
0 0
Fλ(u0)<Fλ and k∇u0kL2(Ω)ku0kL2(Ω) <k∇QkL2(R3)kQkL2(R3).
∗
Then the corresponding solution u to (1.1) is global. Moreover, for all t ∈ R we
have
k∇u(t)kL2(Ω)ku(t)kL2(Ω) <k∇QkL2(R3)kQkL2(R3)
and
1k∇u(t)k2 ≤E(u)≤ 1k∇u(t)k2 .
6 L2(Ω) 2 L2(Ω)
In particular,
Fλ(u)∼ku(t)k2 .
H1
0
Furthermore, with δ >0 such that
Fλ(u )<(1−δ)Fλ,
0
∗