Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains

0.3 MB·English

by  Ramona Anton

GLOBAL EXISTENCE FOR DEFOCUSING CUBIC NLS
AND GROSS-PITAEVSKII EQUATIONS IN THREE
DIMENSIONAL EXTERIOR DOMAINS
RAMONAANTON
7
0
0 Abstract. We prove global wellposedness in the energy space of the
2 defocusing cubic nonlinear Schr¨odinger and Gross-Pitaevskii equations
on the exterior of a non-trapping domain in dimension 3. The main
n
ingredient is a Strichartz estimate obtained combining a semi-classical
a
J Strichartzestimate[4]withasmoothingeffectonexteriordomains[10].

0
1

] 1. Introduction
P
A Let Θ = , Θ R3, a nontrapping obstacle with compact boundary and
h. let Ω = 6∁Θ∅. In⊂this paper we are interested in the Cauchy problem for
t the cubic defocusing NLS equation (here written with Dirichlet boundary
a
m conditions) on Ω :
[ i∂ u+ u = u2u, on R Ω
t
△ | | ×
1 (1)  ut=0 = u0, on Ω
v  u | = 0.
4 |R×∂Ω
0 Thisequationappearsinthenonlinearopticsandmoregenerallyinpropaga-
3
tionofnonlinearwaves. FormoredetailsonnonlinearSchro¨dingerequations
1
see for example the books of C.Sulem-P.L.Sulem [22], T.Cazenave [11] and
0
7 the references therein.
0 There is a wide literature on the Cauchy problem in the Euclidean space.
/
h Oneofthemaintools inaddressingthisproblemistheStrichartz inequality,
at which translates the dispersive property of the linear Schro¨dinger flow. We
m refer to the work of Strichartz [23], Ginibre-Velo [14] and Keel-Tao [19].
: Recently, thequestionoftheinfluenceofthegeometry onthesolution has
v
been studied. Let us mention the work of J.Bourgain [8] on the tori Td for
i
X
d = 2,3 and of N.Burq-P.G´erard-N.Tzvetkov [9], [10] on compact manifold
r and exterior of non-trapping obstacles.
a
In recent works on superfluidity and Bose-Einstein condensates (see for
example the book of A.Aftalion [2]) the following variant of NLS (1) is
studied
i∂ u+ u = (u2 1)u, on R Ω
t
△ | | − ×
(2)  ut=0 = u0, on Ω
 ∂∂uν|R×|∂Ω = 0.
This is called the cubic Gross-Pitaevskii equation with Neumann bound-
ary conditions. The main difference between the NLS (1) and the Gross-
Pitaevskiiequation(2)isintheirenergyspace. ForGross-Pitaevskii itreads
E = u H1 (Ω), u L2(Ω), u2 1 L2(Ω) .
loc
{ ∈ ∇ ∈ | | − ∈ }
1
Namely, the initial datum in the energy space, u E, is not an L2(Ω)
0
∈
function. In [6], [5], [3], [16], [17], [12] the question of existence of travelling
waves and vortices is studied. We are interested here in providing a math-
ematical background for the study of the dynamics of these phenomena.
More precisely, we are interested in showing wellposedness in the energy
space. There have been previous works on the Cauchy problem for the
Gross-Pitaevskii equation : P.E.Zhidkov [24], [25] in Zhidkov spaces X1(R),
F.B´ethuel- J.C.Saut [6] in the space of functions 1+H1(Rd), for d = 2,3,
P.G´erard in [15] in the energy space on the whole Euclidean space Rd, for
d= 2,3,4, C.Gallo [13] in the energy space u +H1(Ω) for exterior domains
0
in d = 2.
For both (1) and (2) the method we use is based on a new Strichartz
estimateobtainedcombiningasmoothingeffectinexteriordomains[10]with
a semiclassical Strichartz estimate on small intervals of time depending on
the frequencies where the flow is localised [4]. In dimension 2 the smoothing
effect [10] provides wellposedness for both NLS [10] and Gross-Pitaevskii
[13], with all power nonlinearities. In dimension 3 the smoothing effect only
provides wellposedness of subcubic nonlinearities [10], [13]. Improving the
Strichartz inequality allows us to treat the cubic nonlinearity in exterior
domains in dimension 3.
Let us recall the definition of an admissible pair.
Definition 1. A pair (p,q) is called admissible in dimension 3 if p 2 and
≥
2 3 3
+ = .
p q 2
The Strichartz inequality we obtain is the following.
Proposition 1.1. For (p,q) an admissible couple in dimension 3 and ǫ > 0,
1 +ǫ
there exists c > 0 such that for all u H2p (Ω),
ǫ 0 ∈ D
(3) ||eit△Du0||Lp(I,Lq(Ω)) ≤ cǫ||u0||H21p+ǫ(Ω)
D
A similar result holds for the linear Schro¨dinger flow with Neumann bound-
ary conditions.
Having a Strichartz inequality we obtain classically a local existence the-
orem for (1) by Picard iteration scheme. These also enables propagation of
the regularity of the initial data. Local existence in the energy space H1(Ω)
0
combined with the conservation of the energy (and for defocusing nonlin-
earity of the H1(Ω) norm) enables us to conclude that the solution to (1) is
0
global in time.
Theorem 1.2. For all u H1(Ω) there exists an unique solution
0 ∈ 0
u C(R,H1(Ω)) Lp (R,L (Ω))
∈ 0 ∩ loc ∞
(for every 2 < p < 3) of equation (1). Moreover, for every T > 0 and
for every bounded subset B of H1(Ω), the flow u u is Lipschitz from B
0 0 7→
to C([ T,T],H1(Ω)). For 1 < σ 2 and u H1(Ω) Hσ(Ω) we have
− 0 ≤ 0 ∈ 0 ∩
u C([ T,T],H1(Ω) Hσ(Ω)).
∈ − 0 ∩
2
For Gross-Pitaevskii equation, as u E is not an L2(Ω) function, the
0
∈
Strichartz inequality does not apply directly. We adapt the arguments of
[15] to the boundary case for the description of the structure of E and of
the action of the linear Schro¨dinger group on E. In particular we define a
structure of complete metric space on E. The global existence theorem for
the Gross-Pitaevskii equation (2) follows by combining the latter structure
with dispersive estimates (3).
Theorem 1.3. For all u E there exists an unique solution
0
∈
u C(R,E) Lp (R,L (Ω))
∈ ∩ loc ∞
(for every 2 < p < 3) of equation (2). Moreover, the following properties
hold: for every bounded subset B of E there exists T > 0 such that for all
u B the flow u u is Lipschitz from B to C([ T,T],E) ; we have
0 0
∈ 7→ −
u u C(R,H1(Ω)) ; if u E is such that u L2(Ω) and ∂u0 = 0,
− L ∈ 0 ∈ △ 0 ∈ ∂ν
then u C(R,L2(Ω)).
△ ∈
Remark 1. After the completion of this work Blair-Smith-Sogge [7] an-
nounced an improved Strichartz inequality on boundary domains. They
prove a Strichartz inequality with a loss of 4 derivatives as opposed to the
3p
Strichartz inequality [4] with a loss of 3 derivatives we used here. Although
2p
this may improve our Strichartz inequality (3) it does not improve the well-
posedness results of Theorem 1.2 and of Theorem 1.3.
The structure of the paper is as follows : in Section 2 we show how we
obtaintheStrichartzestimate(3). InSection3wegivetheproofofTheorem
1.2. In Section 4 we deal with the Gross-Pitaevskii equation (1) and we give
the proof of Theorem 1.3.
Acknowledgments : The author would like to thank P.G´erard for propos-
ing the subject of this article and for many helpfull and stimulating discus-
sions. The result of this article form part of author’s PhD thesis defended
at Universit´e Paris Sud, Orsay, under P.G´erard’s direction.
2. Strichartz estimate in exterior domains
The idea is to combine Strichartz inequality on exterior domains [4] with
the gain of 1 derivative from the smoothing effect [10]. Rather than using
2
the Strichartz estimate with loss of 3 +ǫ derivatives (45) of [4], we prefer
2p
to use the Strichartz estimate without loss of derivatives (Proposition 4.13
of [4]), that holds for frequency localised initial data and small intervals of
time depending on the frequency.
In order to do that here, we need to recall some of the notations and
results from [4]. That is done in Subsection 2.1. In Subsection 2.2 we
recall the results of N.Burq, P.G´erard and N.Tzvetkov [10] concerning the
smoothingeffectandStrichartzestimateawayfromtheobstacle. Subsection
2.3 is the core of this section. We prove a new Strichartz estimate close to
the obstacle by combining semiclassical Strichartz estimate and smoothing
effect. In Subsection 2.4 we deduce the proof of Proposition 1.1.
3
2.1. Preliminaries. We recall here the classical mirror reflection that al-
lows us to pass from a manifold with boundary to a boundaryless manifold.
This method consists in taking a copy of the domain and glue it to the
initial one by identifying the points of the boundary. In order for this to
be a manifold we have to choose the coordinates carefully. Thus, taking
normal coordinates at the boundary is like straightening a neighborhood of
the boundary into a cylinder ∂Ω [0,1) and gluing the two cylinders along
×
the boundary makes a nice smooth manifold. This can be properly done
using for example tubular neighborhoods (e.g. [20], pp. 468 and 74). Let
M = Ω 0 Ω 1 , where we identify (p,0) with (p,1) for p ∂Ω.
∂Ω
×{ }∪ ×{ } ∈
Lemma. ([20])ThereisauniqueC structure onMsuchthat Ω j ֒ M
∞
×{ } →
isC and χ˜ : U 0 U 1 ∂Ω ( 1,1) isa diffeomorphism, where
∞ ∂Ω
×{ }∪ ×{ } → × −
U is a small neighborhood of ∂Ω for which there are deformation retractions
onto ∂Ω.
OnM wedefinethemetricGinducedbythenewcoordinates. Aswehave
chosen coordinates in the normal direction, the metric is well defined over
the boundary, its coefficients are Lipschitz in local coordinates and diagonal
by blocs (no interaction between the normal and the tangent components).
Moreover,
G(r(y)) = G(y),
where r : M M, r(x,0) = (x,1), r2 = Id is the reflection with respect to
→
the boundary ∂Ω.
For the Dirichlet problem we introduce the space H1 of functions of
AS
H1(M) which are anti-symmetric with respect to the boundary. Let
H1 = v :M C, v H1(M), v(y) = v(r(y)) .
AS
{ → ∈ − }
Note that for v H1 the restriction v is in H1(Ω) and every function
fromH1 is obta∈inedASfromafunctionof|ΩH×10(Ω). We0shallprovethestability
AS 0
of H1 under the action of eit G.
AS △
Bycomplex interpolation defineHs for s [0,1] anddeduceits stability
AS ∈
undertheaction ofeit G. Moreover, therestriction toΩoffunctionsinHs
△ AS
belongs to Hs(Ω) and vice versa. This allows us to deduce the Strichartz
D
inequality for eit D on Ω from the Strichartz inequality for eit G on M.
△ △
Similarly, we can define for the Neumann problem the space H1 of
S
symmetric functions with respect to the boundary. This space is also stable
under the action of eit G.
△
H1 = v :M C, v H1(M), v(y) = v(r(y)) .
S
{ → ∈ }
Letus prove the stability of H1 underthe action of eit G. Let v H1
AS △ 0 ∈ AS
andv(t,y) = eit△Gv0. Thenv satisfiestoi∂tv(t,y)+ G(y)v(t,y) = 0,v(0) =
△
v . Letv˜(t,y) = v(t,r(y)).Weshalllookfortheequationverifiedbyv˜. First
0
note that v˜(0) = v and ∂ v˜(t,y) = ∂ v(t,y). As G is diagonal by blocks,
0 t t
−
having no interactions between the normal and tangent components, so is
G 1. Thusin thereisnocrossedterm. Consequently v˜(t,y) =
− G(y) G(r(y))
△ △
v(t,y). We see thus that v˜ satisfies to the linear Schro¨dinger equation
G(y)
△
with initial data v (y). But v(t,y) satisfies the same equations. By
0
− −
4
uniqueness we conclude that
v(t,r(y)) = v(t,y).
−
Moreover, if v (t,y) = u (t,y) for all y Ω, then v(t,y) = u(t,y) for all t
0 0
∈
and for all y Ω, where u(t) = eit Du .
△ 0
∈
We prepare the frequency decomposition. We begin with a partition of
unity on M. Since M is flat outside a compact set, let (U ,κ ) be a
j j j J
covering of the area of M where G = Id. This area is compact, so∈we can
6
choose J of finitecardinal. We have M = U U U , whereU
j J j 1, 2, 1,
and U are two disjoint neighborhood o∪f ∈ , di∪ffeom∞o∪rphe∞to Rd B¯. Le∞t
2,
∞ ∞ \
(χ ) , χ , χ : M [0,1] be a partition of unity subordinated to
j j J 1, 2,
∈ ∞ ∞ →
the previous covering. For all j J let χ˜ : M [0,1] be a C function
j ∞
∈ →
such that χ˜ = 1 on the support of χ and the support of χ˜ is contained
j j j
in U . Similarly we define χ˜ , χ˜ : M [0,1]. Let ϕ C (Rd) be
j 1, 2, 0 ∞
supported in a ball centered at∞origin∞and ϕ→ C (Rd) be su∈pported in an
∞
∈
annulus such that for all λ Rd
∈
(4) ϕ (λ)+ ϕ(2 kλ) = 1.
0 −
kXN
∈
Wedefineafamily ofspectraltruncations: forf C (M)andh (0,1)
∞
∈ ∈
let
(5) J f = (κ ) χ˜ ϕ(hD)(κ 1) (χ f) +F f +F f
h j ∗ j −j ∗ j 1,h, 2,h,
Xj J (cid:16) (cid:17) ∞ ∞
∈
and
(6) J f = (κ ) χ˜ ϕ (D)(κ 1) (χ f) +F f +F f,
0 j ∗ j 0 −j ∗ j 1,0, 2,0,
Xj J (cid:16) (cid:17) ∞ ∞
∈
where denotestheusualpullbackoperationandF f = χ˜ ϕ(hD)χ f(x).
∗ h,
∞ ∞ ∞
The following identity holds
∞
J0f(x)+ J2−kf(x)= f(x).
Xk=0
This will be useful for the Littlewood Paley theory.
We need more regularity on the coefficients of the metric than the Lips-
chitz regularity. Therefore we define a regularized metric G as follows : let
h
ψ be a C (Rd) radially symmetric function with ψ 1 near 0. Let
0∞ ≡
(7) Gh = (κj)∗ χ˜jψ(h12D)(κ−j 1)∗(χjG) .
Xj J (cid:16) (cid:17)
∈
The transformation of G into G does not spoil the symmetry. Note also
h
that G converges uniformly in x to G, and thus, for h sufficiently small,
h
G is positive definite. Therefore, G is still a metric. We present some
h h
properties of metrics G and G .
h
Lemma. The metric G : M M (R) is symmetric, positive definite and
d
→
Lipschitz : there exist c,C,c > 0 such that for all x M
1
∈
cId G(x) CId, ∂G c ,
1
≤ ≤ | | ≤
5
where we have denoted by ∂G the derivatives of the metric in a system of
coordinates. The regularized metric G is a C function that verifies the
h ∞
followings : there exists c,C > 0 and c > 0 for all γ Nd such that
γ
∈
cId G (x) CId, ∂γG (x) c h αmax(γ 1,0).
h h γ − | |−
≤ ≤ | | ≤
We present next a collection of estimates on J . There exist constants
h
c >0 such that, for all h (0,1) :
∈
Jh Lp Lp cp, for all 1 p .
• ||[J||, → ] ≤ c and≤ [J≤,∞ ] c.
• || Ahs△oGneh m||La2y→Ln2ot≤aphply tw||o hde△rivGahti||vHe1s→oLn2G≤(x), the similar state-
mentfor[F , ]onlyholdsfortheH1 L2norm: [F , ]
h G h G H1 L2
c. △ → || △ || → ≤
1
• ||Jh(△Gh −△G)||H1→L2 ≤ ch−2.
We define also a spectral cut-off slightly larger than J . Let ϕ˜ be a C
h ∞
function supported in an annulus such that ϕ˜=1 on a neighborhood of the
support of ϕ. We define J˜ just like J , replacing ϕ par ϕ˜ in (5) :
h h
(8) J˜ f = (κ ) χ˜ ϕ˜(hD)(κ 1) (χ f) +F˜ f +F˜ f
h j ∗ j −j ∗ j 1,h, 2,h,
Xj J (cid:16) (cid:17) ∞ ∞
∈
Then the action of J˜ on J and [J , ] is close to identity in Lp Lp
h h h △Gh →
norm, p 2, and L2 L2 norm respectively.
≥ →
J˜hJh Jh Lp Lp cNhN,
• ||[J , − ] || [J→, ≤ ]J˜ c hN, for all N N.
• || h △Gh − h △Gh h||L2→L2 ≤ N ∈
Let us recall also the Strichartz estimate we use from [4].
Lemma. (4.13 of [4]) For all couples (p,q) admissible in dimension 3 and
3
Ih an interval of time such that Ih = ch2, we have
| |
(9) ||Jh∗eit△Ghu0||Lp(Ih,Lq(M)) ≤ c||u0||L2
We prefer to go back to the estimate on eit△Gh since the form of the
Strichartz estimate for eit G is more difficult to handle ((45) of [4]) :
△
||Jh∗eit△Gu0||Lp(Ih,Lq(M)) ≤ ch||u0||H1.
This is due to the fact that and are not both selfadjoint in the
△G △Gh
same space, because of the the volume density 1 .
√detG(x)
2.2. Smoothingeffect andStrichartzestimate awayfromthe obsta-
cle. InthissectionwerecalltworesultsofN.Burq,P.G´erardandN.Tzvetkov
[10]onthesmoothingeffectfortheSchro¨dingerflowonexteriordomainsand
the Strichartz estimate away from the obstacle. The smoothing effect was
obtained via resolvent bounds. For Strichartz estimate they used a strategy
inspiredby G.Staffilani and D.Tataru’s paper[26] on C2 short range pertur-
bation of the free Laplacian on Rd. Thus, they proved that away from the
obstacle the linear Schro¨dinger flow satisfies the usual Strichartz estimates.
We present an equivalent statement on the double manifold.
6
Proposition. (2.7 of [10]) Assume that Θ = . Then for every T > 0, for
every χ C (Rd), d 2, 6 ∅
∈ 0∞ ≥
||χeit△Du0||L2Hs+21(Ω) ≤ c||u0||L2THDs(Ω), for s ∈ [0,1].
T D
Proposition. (2.10 of [10]) For every T > 0, for every χ C (Rd), χ = 1
∈ 0∞
close to Θ, there exists C > 0 such that
(10) ||(1−χ)u||LpTWs,q(Ω) ≤ C||u0||HDs (Ω),
where s [0,1], u(t)= eit Du and (p,q) any Strichartz admissible pair.
△ 0
∈
The proof relies on the use of the smoothing effect and the fact that
(1 χ)eit Du can be seen as a solution to some nonlinear Schro¨dinger
△ 0
equ−ation on Rd.
Although the properties are written for the Dirichlet Laplacian, Remark
1.2 of [10] ensures that the results hold for the Neumann conditions as well.
From theway weconstructed thedoublemanifoldandflows,wededucethat
those results extend easily on the double manifold.
Proposition 2.1. Assume that Θ = . Then for every T > 0, for every
6 ∅
χ C (M),
∈ 0∞
||χeit△Gu0||L2Hs+12(M) ≤ c||u0||L2THs(M), for s ∈ [0,1].
T
Proposition 2.2. For every T > 0, for every χ C (M), χ = 1 close to
∈ 0∞
Θ, where Θ represents the double of Θ, there exists C > 0 such that
D D
(11) ||(1−χ)eit△Gu0||LpTWs,q(M) ≤ C||u0||Hs(M),
where s [0,1] and (p,q) any Strichartz admissible pair.
∈
2.3. Strichartzestimateneartheobstacle. WewanttocombineStrichartz
estimate on domains [4] with smoothing effect [10]. For this we use the
Strichartz estimate of the frequency localised linear flow, without loss of
derivatives, which holds on a small interval of time (see estimate (9) from
Subsection 2.1).
Let ϕ C (R) such that ϕ 1 on [ 1,1], 0 ϕ 1, suppϕ [ 1,1]
∈ 0∞ ≡ −4 4 ≤ ≤ ⊂ −2 2
and there exists J R a discrete set such that ϕ2(t t ) = 1 for all
⊂ t0 J − 0
t R. Let δ > 0 be a small number. If we consPider∈J¯= [ δ,1+δ] ch32J
∈ − ∩
then, for t [ δ,1+ δ],
∈ −2 2
t t
(12) ϕ2 − 0 = 1.
tX0 J¯ (cid:18) ch32 (cid:19)
∈
3 3 3 3
Let us denote by I (t ) = [t ch2,t + ch2], I (t ) = [t ch2,t + ch2],
h 0 0 − 4 0 4 h′ 0 0 − 2 0 2
uL(t) = eit△Gu0 and by
t t
(13) v(t) = ϕ(cid:18) c−h230(cid:19)Jh∗χeit△Gu0.
Notice that v(t) = Jh∗χeit△Gu0 for t ∈ Ih(t0) and supptv ⊂ Ih′. We write the
end-point Strichartz estimate for v on I . Notice that the couple (2,6) is
h′
admissible in dimension 3.
7
Lemma 2.3. For t R, χ˜ C (M) such that χ˜χ = χ and J˜ a spectral
0 ∈ ∈ 0∞ h
cut-off slightly larger than J (8), we have
h
t t
||ϕ(cid:18) c−h320(cid:19)Jh∗χeit△Gu0||L2(Ih′,L6(M)) ≤ch−34||Jh∗χuL||L2(Ih′,L2(M)) +
(14) h34 J˜h∗χ˜uL L2(I′,H1(M)) +c χ˜uL L2(I′,H1(M)).
|| || h || || h
3
Proof. For simplicity, let us suppose that Ih′(t0) = [0,T], where T = ch2.
Then v(t) verifies, for t I (t ), the equation
∈ h′ 0
i∂ v+ v = f +f +f
t △Gh 1 2 3
(cid:26) v = 0
t=0
|
where f = i ϕ t t0 J χu , f = ϕ t t0 [ ,J χ]u and f =
1 ch23 ′(cid:16)c−h23 (cid:17) h∗ L 2 (cid:16)c−h23 (cid:17) △Gh h∗ L 3
ϕ t t0 J χ( )u . BytheDuhamelformulaandusingthatJ˜ J =
(cid:16)c−h23 (cid:17) h∗ △G−△Gh L h∗ h∗
J +c hN in Lp Lp norm, for p 2, we have
h∗ N → ≥
v(t) = v (t)+v (t)+v (t)+c hN,
1 2 3 N
where we define
v1(t) = 13 0tJ˜h∗ei(t−τ)△Ghϕ′ τ−t30 Jh∗χuL(τ)dτ,
ch2 (cid:16)ch2 (cid:17)
v2(t) = −i 0RtJ˜h∗ei(t−τ)△Ghϕ(cid:16)τc−ht320(cid:17)[△Gh,Jh∗χ]uL(τ)dτ,
v3(t) = −iR0tJ˜h∗ei(t−τ)△Ghϕ(cid:16)τc−ht320(cid:17)Jh∗χ(△G −△Gh)uL(τ)dτ.
R
By Minkowski inequality and estimate (9), we have
||v1||L2t(Ih′,L6(M)) ≤ ch−23 0T |ϕ′(cid:16)τc−ht230(cid:17)|||1τ<tJ˜h∗ei(t−τ)△GhJh∗χuL(τ)||L2t(L6x(M))dτ,
≤ ch−32 R0T |ϕ′(cid:16)τc−ht230(cid:17)|||Jh∗χuL(τ)||L2x(M)dτ.
R
3
Using Cauchy Schwarz inequality and ||ϕ′(ch·32)||L2 = ch4, we obtain
3
(15) ||v1||L2t(Ih′,L6(M)) ≤ ch−4||Jh∗χuL||L2(Ih′×M)
3
Similarly,wehave||v2||L2t(Ih′,L6(M)) ≤ ch4||[△Gh,Jh∗χ]uL||L2(Ih′×M).Usingthat
[ ,J χ] c and [ ,J χ] [ ,J χ]J˜ χ˜
|m|△odGuhlo ch∗h |N|H,1→weL2ob≤tain ||△Gh h∗ ||H1→L2 ∼ ||△Gh h∗ h∗ ||H1→L2
−
(16) ||v2||L2t(Ih′,L6(M)) ≤ ch34||J˜h∗χ˜uL||L2(Ih′,H1(M)).
Weestimatethethirdtermv inL2(I ,L6(M))norminasimilarmanner.
3 h′
3
We get : ||v3||L2t(Ih′,L6(M)) ≤ ch4||Jh∗χ(△G − △Gh)uL||L2(Ih′×M). Using the
1
estimate ||Jh∗χ(△G−△Gh)f||L2(M) ≤ ch−2||χ˜f||H1(M), we obtain
1
(17) ||v3||L2t(Ih′,L6(M)) ≤ ch4||χ˜uL||L2(Ih′,H1(M)).
Recalling that v(t) = v (t) + v (t) + v (t), the result follows from the
1 2 3
triangle inequality and the sum of (15), (16) and (17). (cid:3)
8
Weproceed tothesummationover theintervals of timein ordertoobtain
a Strichartz inequality (for the frequency localized flow) on a fixed interval
of time. Letus denotebyI = [0,1] and byI = I+[ δ,δ], whereδ is chosen
δ
−
like in (12).
Lemma 2.4. Under the same notations as in Lemma 2.3, we have
3
||Jh∗χuL||L2(I,L6(M)) ≤ ch−4||Jh∗χuL||L2(Iδ,L2(M)) +
(18) h34||J˜h∗χ˜uL||L2(Iδ,H1(M))+c||χ˜uL||L2(Iδ,H1(M)).
Proof. We sum the square of (14) over t J¯, where J¯ was defined for
0
∈
the identity (12). From (12) and the definition of ϕ we deduce that the
reunion of intervals I (t ), for t J¯, recovers I at most twice. Thus,
h′ 0 0 ∈ δ
f 2 2 f 2 . The result follows by merely observing
Pthat0t∈J¯J|| χ||uL2(Ih′) ≤ || ||L2([−Jδ,1+χδu]) . (cid:3)
|| h∗ L||L2(I,L6(M)) ≤ || h∗ L||L2(Iδ,L6(M))
From (18) we get the Strichartz inequality near the obstacle by means of
Littlewood Paley summation.
Proposition 2.5. For every ǫ > 0 there exists c > 0 such that for (p,q)
ǫ
admissible in dimension 3,
(19) ||χeit△Gu0||Lp([0,1],W21p−ǫ,q(M)) ≤ cǫ||u0||H1p(M).
Proof. We apply a corollary of the Littlewood Paley theorem for p,q 2 :
≥
1
2
∞
(20) ||u||LpT(Wσ,q) ≤ c||S0u||Lp(Lq)+Xj=022jσ||△ju||2Lp(Lq)
Here we apply it for (p,q) = (2,6) and h = 2 j, = J , σ= 1 ǫ > 0
− △j 2∗−j 4 −
andu= χu onI = [0,1]. Theleft handsidetermreads χu .
L L L2(I,Wσ,6(M))
|| ||
Using (18), the parenthesis from the right hand side term is bounded by a
sum ∞j=0 of terms like
P
2j(2 2ǫ) χu 2 +2 j(1+2ǫ) χu 2 +2 2jǫ χ˜u 2 .
− ||△j L||L2(Iδ,L2(M)) − ||△j L||L2(Iδ,H1(M)) − || L||L2(Iδ,H1(M))
Using the Plancherel theorem for the first two series and the geometric
summation for the third (notice that ∞j=02−2jǫ = cǫ), we obtain
P
||χuL||L2(I,Wσ,6(M)) ≤ ||χuL||L2(Iδ,H1−ǫ(M))+||χuL||L2(Iδ,H21−ǫ(M))+||χ˜uL||L2(Iδ,H1(M)).
We apply the smoothing effect (see Proposition 2.7 of [10] and the trans-
lation onto the double). Thus,
χu c u .
|| L||L2(I,W41−ǫ,6(M)) ≤ || 0||H12(M)
We want to perform a complex interpolation between the previous estimate
and the conservation of the L2 norm (we used also 0 χ 1) :
≤ ≤
χuL L∞(I,L2(M)) c u0 L2(M).
|| || ≤ || ||
Using a weight of 2, respectively 1 2, we get an estimates of Strichartz
p − p
type with loss of derivatives :
χu c u ,
|| L||Lp(I,W21p−2ǫ,q(M)) ≤ || 0||Hp1(M)
9
where (p,q) satisfy 2 + 3 = 3, i.e. they form an admissible couple in
p q 2
dimension 3. (cid:3)
2.4. Proof of Proposition 1.1. Combining estimates (19) (Strichartz es-
timateneartheboundaryofΩ)with(11)(Strichartzestimateaway fromthe
boundary) for s = 1 ǫ, we obtain, using that v v ,
2p − || 0||H21p−ǫ(M) ≤ || 0||Hp1(M)
||eit△Gv0||Lp([0,1],W21p−ǫ,q(M)) ≤ cǫ||v0||Hp1(M).
1 +ǫ 1 +ǫ
Let u H2p (Ω) and let v H2p (Ω) be such that v = u . By
uniquene0s∈s andD stability at refle0xi∈on oAvSer the boundary of Ω0|oΩf the 0linear
flow (see Section 2.1), we have eit Gv = eit Du . Thus,
△ 0 Ω △ 0
|
eit Gv eit Du
△ 0 Lp([0,1],Ws,q(M)) △ 0 Lp([0,1],Ws,q(Ω))
|| || ≈ || ||
and v u . We obtain,
0 Hs(M) 0 Hs(Ω)
|| || ≈ || ||
||eit△Du0||Lp([0,1],W21p−ǫ,q(Ω)) ≤ cǫ||u0||Hp1(Ω).
We apply the ellipticity of the Laplacian to deduce a whole range
D
of Strichartz inequalities : let u˜0 = (1−△D)△−σ2u0, where σ = 21p −ǫ−s,
s [0,1]. If u Hs0(Ω) then u˜ Hs0 σ(Ω). We obtain the following
0 0 −
∈ ∈ ∈
inequality for eit Du :
△ 0
(21) ||eit△Du0||Lp([0,1],Ws,q(Ω)) ≤ cǫ||u0||Hs+21p+ǫ(Ω).
1 +ǫ 1 +ǫ
For u H2p (Ω), we consider v H2p (M) be such that v = u .
We dedu0c∈e asNabove the Strichartz in0eq∈ualiSty for the linear Schro¨din0|gΩer flow0
with Neumann Laplacian.
3. Global existence for NLS
Having a Strichartz inequality we obtain classically a local existence the-
orem by Picard iteration scheme. These also enables propagation of the
regularity of the initial data. Local existence in the energy space H1(Ω)
0
combined with the conservation of the energy (and for defocusing nonlin-
earity of the H1(Ω) norm) enables us to conclude that the solution to is
0
global in time.
Proof. ( of Theorem 1.2) Let us denote by X = C([ T,T],H1(Ω))
T − 0 ∩
Lp([ T,T],L (Ω)) and, for a fix u B H1(Ω), by Φ : X X
− ∞ 0 ∈ ⊂ 0 T → T
the functional
t
Φ(u)(t) = eit u i ei(t τ) u(τ)2u(τ)dτ.
△ 0 − △
− Z | |
0
The space X is a complete Banach space for the following norm
T
||u||XT = max|t|≤T||u(t)||H1(Ω) +||u||Lp([−T,T],L∞(Ω)).
We prove that for a T > 0 and R >0 small enough, Φ is a contraction from
B(0,R) X into itself. We begin by estimating the H1 norm of Φ(u) :
T
⊂
||Φ(u)(t)||H1 ≤ ||u0||H1 +cT1−p2||u||2Lp(L∞)||u||L∞T (H1) ≤ ||u0||H1 +cT1−p2||u||3XT.
10