Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3

0.95 MB·English

by  Jim Colliander

GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE
ENERGY-CRITICAL NONLINEAR SCHRO¨DINGER EQUATION
6 IN R3
0
0
2 J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO

n
a
J Abstract. Weobtainglobalwell-posedness,scattering,andglobalL1t,0xspace-
time bounds for energy-class solutions to the quintic defocusing Schro¨dinger
8 equationinR1+3,whichisenergy-critical. Inparticular,thisestablishesglobal

existence of classical solutions. Our work extends the results of Bourgain [4]
]
P and Grillakis [20], which handled the radial case. The method is similar in
spirittotheinduction-on-energystrategyofBourgain[4],butweperformthe
A
inductionanalysisinbothfrequencyspaceandphysicalspacesimultaneously,
h. and replace the Morawetz inequality by an interaction variant (first used in
t [12], [13]). The principal advantage of the interaction Morawetz estimate is
a that it is not localized to the spatial origin and so is better able to handle
m nonradial solutions. Inparticular, this interaction estimate, together withan
almost-conservation argument controlling the movement of L2 mass in fre-
[
quency space,rulesoutthepossibilityofenergyconcentration.

7
v
9
2
1 Contents
2
0 1. Introduction 2
4 1.1. Critical NLS and main result 2
0
1.2. Notation 7
/
h 2. Local conservation laws 9
t 3. Review of Strichartz theory in R1+3 11
a
m 3.1. Linear Strichartz estimates 11
3.2. Bilinear Strichartz estimate 14
:
v 3.3. Quintilinear Strichartz estimates 16
i
X 3.4. Local well-posedness and perturbation theory 19
4. Overview of proof of global spacetime bounds 23
r
a 4.1. Zeroth stage: Induction on energy 23
4.2. First stage: Localization control on u 26
Date:28January2004.
1991 Mathematics Subject Classification. 35Q55.
Key words and phrases. NonlinearSchro¨dingerequation, well-posedness.
J.C.issupportedinpartbyN.S.F.GrantDMS0100595,N.S.E.R.C.GrantR.G.P.I.N.250233-
03andtheSloanFoundation.
M.K.was supportedinpartbyN.S.F.GrantDMS-0303704; andbythe McKnightandSloan
Foundations.
G.S.issupportedinpartbyN.S.F.GrantDMS-0100375,N.S.F.GrantDMS-0111298through
theIAS,andtheSloanFoundation.
H.T. is supported in part by J.S.P.S. Grant No. 15740090 and by a J.S.P.S. Postdoctoral
FellowshipforResearchAbroad.
T.T.isaClayPrizeFellowandissupportedinpartbygrants fromthePackardFoundation.
1
2 J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO
4.3. Second stage: Localized Morawetz estimate 30
4.4. Third stage: Nonconcentration of energy 33
5. Frequency delocalized at one time =⇒ spacetime bounded 34
6. Small L6 norm at one time =⇒ spacetime bounded 37
x
7. Spatial concentration of energy at every time 39
8. Spatial delocalized at one time =⇒ spacetime bounded 40
9. Reverse Sobolev inequality 46
10. Interaction Morawetz: generalities 47
10.1. Virial-type Identity 47
10.2. Interaction virial identity and general interaction Morawetz estimate for general equations 49
11. Interaction Morawetz: The setup and an averaging argument 52
12. Interaction Morawetz: Strichartz control 61
13. Interaction Morawetz: Error estimates 64
14. Interaction Morawetz: A double Duhamel trick 67
15. Preventing energy evacuation 71
15.1. The setup and contradiction argument 71
15.2. Spacetime estimates for high, medium, and low frequencies 73
15.3. Controlling the localized L2 mass increment 77
16. Remarks 81
References 83
1. Introduction
1.1. Critical NLS and main result. We consider the Cauchy problem for the
quintic defocusing Schr¨odinger equation in R1+3
iu +∆u=|u|4u
(1.1) t
u(0,x)=u (x).
0
(cid:26)
where u(t,x) is a complex-valued field in spacetime R ×R3. This equation has as
t x
Hamiltonian,
1 1
(1.2) E(u(t)):= |∇u(t,x)|2+ |u(t,x)|6 dx.
2 6
Z
Since the Hamiltonian (1.2) is preservedby the flow (1.1) we shall often refer to it
as the energy and write E(u) for E(u(t)).
SemilinearSchr¨odingerequations-withandwithoutpotentials,andwithvarious
nonlinearities - arise as models for diverse physical phenomena, including Bose-
Einstein condensates [23, 35] and as a description of the envelope dynamics of a
general dispersive wave in a weakly nonlinear medium (see e.g. the survey in [43],
Chapter 1). Our interesthere in the defocusing quintic equation(1.1) is motivated
mainly though by the fact that the problem is critical with respect to the energy
norm. Specifically, we map a solution to another solution through the scaling
u7→uλ defined by
1 t x
(1.3) uλ(t,x):= u( , ),
λ1/2 λ2 λ
and this scaling leaves both terms in the energy invariant.
The Cauchy problem for this equation has been intensively studied ([9], [20],
[4], [5],[18], [26]). It is known (see e.g. [10, 9]) that if the initial data u (x) has
0
SCATTERING FOR 3D CRITICAL NLS 3
finiteenergy,thentheCauchyproblemislocallywell-posed,inthesensethatthere
exists a local-in-time solution to (1.1) which lies in C0H˙1 ∩L10, and is unique
t x t,x
in this class; furthermore the map from initial data to solution is locally Lipschitz
continuousinthesenorms. Iftheenergyissmall,thenthesolutionisknowntoexist
globally in time, and scatters to a solution u (t) to the free Schr¨odinger equation
±
(i∂ +∆)u = 0, in the sense that ku(t)−u (t)k → 0 as t → ±∞. For
t ± ± H˙1(R3)
(1.1) with large initial data, the arguments in [10, 9] do not extend to yield global
well-posedness,even with the conservationof the energy (1.2), because the time of
existence given by the local theory depends on the profile of the data as well as on
theenergy1. Forlargefiniteenergydatawhichisassumedtobeinadditionradially
symmetric,Bourgain[4]provedglobalexistenceandscatteringfor(1.1)inH˙1(R3).
Subsequently Grillakis [20] gave a different argument which recovered part of [4] -
namely, global existence from smooth, radial, finite energy data. For general large
data - in particular, general smooth data - global existence and scattering were
open.
Our main result is the following global well-posedness result for (1.1) in the
energy class.
Theorem 1.1. For any u with finite energy, E(u ) < ∞, there exists a unique2
0 0
global solution u∈C0(H˙1)∩L10 to (1.1) such that
t x t,x
∞
(1.4) |u(t,x)|10 dxdt≤C(E(u )).
0
Z−∞ZR3
for some constant C(E(u )) that depends only on the energy.
0
Asiswell-known(seee.g. [5],or[13]forthesub-criticalanalogue),theL10 bound
t,x
above also gives scattering, asymptotic completeness, and uniform regularity:
Corollary 1.2. Let u have finite energy. Then there exists finite energy solutions
0
u (t,x) to the free Schro¨dinger equation (i∂ +∆)u =0 such that
± t ±
ku (t)−u(t)k →0 as t→±∞.
± H˙1
Furthermore, the maps u 7→u (0) are homeomorphisms from H˙1(R3) to H˙1(R3).
0 ±
Finally, if u ∈Hs for some s>1, then u(t)∈Hs for all time t, and one has the
0
uniform bounds
supku(t)k ≤C(E(u ),s)ku k .
Hs 0 0 Hs
t∈R
ItisalsofairlystandardtoshowthattheL10 bound(1.4)impliesfurtherspace-
t,x
time integrabilityonu, for instance u obeys allthe Strichartz estimates that a free
solution with the same regularity does (see, for example, Lemma 3.12 below).
The results here have analogs in previous work on second order wave equations
onR3+1 with energy-critical(quintic) defocusing nonlinearities. Globalintime ex-
istence for such equations from smooth data was shown by Grillakis [21],[22] (for
radial data see Struwe [42], for small energy data see Rauch [36]); global in time
solutions from finite energy data was shownin Kapitanski[25], Shatah-Struwe[39].
1Thisisinconstrast withsub-criticalequations suchasthe cubicequation iut+∆u=|u|2u,
forwhichonecanusethelocalwell-posednesstheorytoyieldglobalwell-posednessandscattering
evenforlargeenergydata(see[17],andthesurveys[7],[8]).
2Infact,uniquenessactuallyholdsinthelargerspaceC0(H˙1)(thuseliminatingtheconstraint
t x
thatu∈L10),asonecanshowbyadaptingtheargumentsof[27],[14],[15];seeSection16.
t,x
4 J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO
For an analog of the scattering statement in Corollary 1.2 for the critical wave
equation see Bahouri-Shatah [2], Bahouri-G´erard [1]; for the scattering statement
for Klein-Gordon equations see Nakanishi [30] (for radial data, see Ginibre-Soffer-
Velo[16]). The existence results mentioned here all involve an argument showing
thatthesolution’senergycan’tconcentrate. Theseenergynonconcentrationproofs
combineMorawetz inequalities (a priori estimatesfor the nonlinearequationwhich
boundsomequantitythatscaleslikeenergy)withcarefulanalysisthatstrengthens
the Morawetzbound to controlof energy. Besides the presence of infinite propaga-
tionspeeds,amaindifferencebetween(1.1)andthehyperbolicanalogsisthathere
time scales like λ2, and as a consequence the quantity bounded by the Morawetz
estimate is supercritical with respect to energy.
Section4belowprovidesafairlycompleteoutlineoftheproofofTheorem1.1. In
this introductionweonlybrieflysketchsomeofthe ideasinvolved: asuitable mod-
ificationoftheMorawetzinequalityfor(1.1),alongwiththefrequencylocalizedL2
almost-conservationlaw that we’llultimately use to prohibitenergyconcentration.
A typical example of a Morawetz inequality for (1.1) is the following bound due
to Lin-Strauss [33] who cite [34] as motivation,
|u(t,x)|6
(1.5) dxdt.(supku(t)k )2
ZIZR3 |x| t∈I H˙1/2
for arbitrary time intervals I. (The estimate (1.5) follows from a computation
showing the quantity,
x
(1.6) Im(u¯∇u· )dx
R3 |x|
Z
is monotone in time.) Observe that the right-hand side of (1.5) will not grow in I
if the H1 and L2 norms are bounded, and so this estimate gives a uniform bound
on the left hand side where I is any interval on which we know the solution exists.
However, in the energy-critical problem (1.1) there are two drawbacks with this
estimate. The first is that the right-hand side involves the H˙1/2 norm, instead of
the energy E. This is troublesome since any Sobolev norm rougher than H˙1 is
supercritical with respect to the scaling (1.3). Specifically, the right hand side of
(1.5) increases without bound when we simply scale givenfinite energy initial data
according to (1.3) with λ large. The second difficulty is that the left-hand side
is localized near the spatial origin x = 0 and does not convey as much informa-
tion about the solution u away from this origin. To get around the first difficulty
Bourgain[4]andGrillakis[20]introducedalocalizedvariantoftheaboveestimate:
|u(t,x)|6
(1.7) dxdt.E(u)|I|1/2.
|x|
ZIZ|x|.|I|1/2
As an example of the usefulness of (1.7), we observe that this estimate prohibits
the existence of finite energy (stationary) pseudosoliton solutions to (1.1). By a
(stationary) pseudosoliton we mean a solution such that |u(t,x)| ∼ 1 for all t ∈ R
and |x| . 1; this notion includes soliton and breather type solutions. Indeed,
applying (1.7) to such a solution, we would see that the left-hand side grows by at
least |I|, while the right-hand side is O(|I|12), and so a pseudosoliton solution will
lead to a contradiction for |I| sufficiently large. A similar argument allows one to
use (1.7) to prevent “sufficiently rapid” concentration of (potential) energy at the
SCATTERING FOR 3D CRITICAL NLS 5
origin; for instance, (1.7) can also be used to rule out self-similar type blowup3,
where the potential energy density |u|6 concentrates in the ball |x| < A|t−t | as
0
t→t− forsomefixedA>0. In[4],onemainuseof (1.7)wastoshowthatforeach
0
fixed time interval I, there exists at least one time t ∈ I for which the potential
0
energy was dispersed at scale |I|1/2 or greater (i.e. the potential energy could not
concentrate on a ball |x|≪|I|1/2 for all times in I).
To summarize, the localizedMorawetz estimate (1.7) is very good at preventing
u from concentrating near the origin; this is especially useful in the case of radial
solutions u, since the radial symmetry (combined with conservation of energy)
enforces decay of u awayfromthe origin,and so resolvesthe second difficulty with
theMorawetzestimatementionedearlier. However,theestimateislessusefulwhen
the solutionis allowedto concentrateawayfromthe origin. For instance,ifwe aim
to preclude the existence of a moving pseudosoliton solution, in which |u(t,x)|∼1
when |x−vt| . 1 for some fixed velocity v, then the left-hand side of (1.7) only
grows like log|I| and so one does not necessarily obtain a contradiction4.
Itisthusofinteresttoremovethe1/|x|denominatorin(1.5),(1.7),sothatthese
estimatescanmoreeasilypreventconcentrationatarbitrarylocationsinspacetime.
In [12], [13] this was achieved by translating the origin in the integrand of (1.6)
to an arbitrary point y, and averaging against the L1 mass density |u(y)|2 dy. In
particular, the following interaction Morawetz estimate5
(1.8) |u(t,x)|4 dxdt.ku(0)k2 (supku(t)k )2
L2 H˙1/2
ZIZR3 t∈I
was obtained. (We have since learned that this averaging argument has an analog
in early work presenting and analyzing interaction functionals for one dimensional
hyperbolic systems, e.g. [19, 38].) This L4 estimate already gives a short proof
t,x
of scattering in the energy class (and below!) for the cubic nonlinear Schr¨odinger
equation (see [12], [13]); however, like (1.5), this estimate is not suitable for the
critical problem because the right-hand side is not controlled by the energy E(u).
One could attempt to localize (1.8) as in (1.7), obtaining for instance a scale-
invariant estimate such as
(1.9) |u(t,x)|4 dxdt.E(u)2|I|3/2
ZIZ|x|.|I|1/2
but this estimate, while true (in fact it follows immediately from Sobolev and
Ho¨lder),is uselessfor suchpurposesasprohibiting soliton-likebehaviour,since the
3This is not the only type of self-similar blowup scenario; another type is when the energy
concentrates in a ball |x| ≤ A|t−t0|1/2 as t → t−0. This type of blowup is consistent with the
scaling(1.3)andisnotdirectlyruledoutby(1.7),howeveritcaninsteadberuledoutbyspatially
localmassconservationestimates. See[4],[20].
4Atfirstglanceitmayappearthattheglobalestimate(1.5)isstillabletoprecludetheexistence
of such a pseudosoliton, since the right-hand side does not seem to grow much as I gets larger.
This can be done in the cubic problem (see e.g. [17]) but in the critical problem one can lose
controloftheH˙1/2 norm,byaddingsomeverylowfrequencycomponents tothesolitonsolution
u. Onemightobject that one coulduse L2 conservation tocontrol theH1/2 norm,however one
canrescalethesolutiontomaketheL2 norm(andhence theH˙1/2 norm)arbitrarilylarge.
5Strictly speaking, in [12], [13] this estimate was obtained for the cubic defocusing nonlinear
Schro¨dinger equation instead of the quintic, but the argument in fact works for all nonlinear
Schro¨dinger equations with a pure power defocusing nonlinearity, and even for a slightly more
generalclassofrepulsivenonlinearitiessatisfyingastandardmonotonicitycondition. See[13]and
Section10belowformorediscussion.
6 J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO
left-hand side grows like |I| while the right-hand side grows like |I|3/2. Nor is this
estimate useful for preventing any sort of energy concentration.
Our solution to these difficulties proceeds in the context of an induction-on-
energy argument as in [4]: assume for contradiction that Theorem 1.1 is false,
and consider a solution of minimal energy among all those solutions with L10
x,t
norm above some threshhold. We first show, without relying on any of the above
Morawetz-typeinequalities,thatsuchaminimal energyblowupsolutionwouldhave
to be localized in both frequency and in space at all times. Second, we prove that
this localized blowup solution satisfies Proposition4.9, which localizes (1.8) in fre-
quency rather than in space. Roughly speaking, the frequency localized Morawetz
inequality of Proposition 4.9 states that after throwing away some small energy,
low frequency portions of the blow-up solution, the remainder obeys good L4 es-
t,x
timates. In principle, this estimate should follow simply by repeating the proof
of (1.8) with u replaced by the high frequency portion of the solution, and then
controllingerrorterms;howeversomeoftheerrortermsareratherdifficultandthe
proofofthe frequency-localizedMorawetzinequalityisquite technical. We empha-
size that, unlike the estimates (1.5), (1.7), (1.8), the frequency-localized Morawetz
inequality (4.19) is not an a priori estimate valid for all solutions of (1.1), but
instead applies only to minimal energy blowup solutions; see Section 4 for further
discussion and precise definitions.
The strategy is then to try to use Sobolev embedding to boost this L4 control
t,x
toL10 controlwhichwouldcontradicttheexistenceoftheblow-upsolution. There
t,x
is,however,aremainingenemy,whichisthatthesolutionmayshiftitsenergyfrom
low frequencies to high, possibly causing the L10 norm to blow up while the L4
t,x t,x
normstays bounded. To preventthis we look at what such a frequency evacuation
would imply for the location - in frequency space - of the blow-up solution’s L2
mass. Specifically, we prove a frequency localized L2 mass estimate that gives us
information for longer time intervals than seems to be available from the spatially
localized mass conservation laws used in the previous radial work ([4, 20]). By
combining this frequency localized mass estimate with the L4 bound and plenty
t,x
of Strichartz estimate analysis, we can control the movement of energy and mass
from one frequency range to another, and prevent the low-to-high cascade from
occurring. The argument here is motivated by our previous low-regularity work
involving almost conservation laws (e.g. [13]).
The remainder of the paper is organized as follows: Section 2 reviews some
simple, classical conservation laws for Schr¨odinger equations which will be used
througout, but especially in proving the frequency localized interaction Morawetz
estimate. In section 3 we recall some linear and multilinear Strichartz estimates,
along with the useful nonlinear perturbation statement of Lemma 3.10. Section 4
outlines in some detail the argumentbehind our main Theorem,leaving the proofs
ofeachsteptosections 5- 15ofthe paper. Section16presentssome miscellaneous
remarks, including a proof of the unconditional uniqueness statement alluded to
above.
Acknowledgements:
We thank the Institute for Mathematics and its Applications (IMA) for host-
ing our collaborative meeting in July 2002. We thank Andrew Hassell, Sergiu
Klainerman, and Jalal Shatah for interesting discussions related to the interaction
Morawetzestimate, and Jean Bourgainfor valuable comments on an early draft of
SCATTERING FOR 3D CRITICAL NLS 7
this paper, to Monica Visan and the anonymous referee for their thorough reading
ofthemanuscriptandformanyimportantcorrections,andtoChangxingMiaoand
GuixiangXuforfurthercorrections. WethankManoussosGrillakisforexplanatory
detailsrelatedto[20]. Finally,itwillbecleartothereaderthatourworkhererelies
heavily in places on arguments developed by J. Bourgain in [4].
1.2. Notation. IfX,Y arenonnegativequantities,weuseX .Y orX =O(Y)to
denotethe estimate X ≤CY forsomeC (whichmaydependonthe criticalenergy
E (see Section 4) but not on any other parameter such as η), and X ∼ Y to
crit
denote the estimate X .Y .X. We use X ≪Y to mean X ≤cY for some small
constant c (which is again allowed to depend on E ).
crit
We use C ≫1 to denote various large finite constants, and 0<c≪1 to denote
various small constants.
The Fourier transform on R3 is defined by
fˆ(ξ):= e−2πix·ξf(x) dx,
R3
Z
giving rise to the fractional differentiation operators|∇|s, h∇is defined by
|\∇|sf(ξ):=|ξ|sfˆ(ξ); h\∇isf(ξ):=hξisfˆ(ξ)
wherehξi:=(1+|ξ|2)1/2. Inparticular,wewilluse∇todenotethespatialgradient
∇ . This in turn defines the Sobolev norms
x
kfkH˙s(R3) :=k|∇|sfkL2(R3); kfkHs(R3) :=kh∇isfkL2(R3).
More generally we define
kfkW˙ s,p(R3) :=k|∇|sfkLp(R3); kfkWs,p(R3) :=kh∇isfkLp(R3)
for s∈R and 1<p<∞.
Weleteit∆bethefreeSchr¨odingerpropagator;intermsoftheFouriertransform,
this is given by
(1.10) \eit∆f(ξ)=e−4π2it|ξ|2fˆ(ξ)
while in physical space we have
(1.11) eit∆f(x)= 1 ei|x−y|2/4tf(y) dy
(4πit)3/2 R3
Z
for t 6= 0, using an appropriate branch cut to define the complex square root. In
particularthepropagatorpreservesalltheSobolevnormsHs(R3)andH˙s(R3),and
also obeys the dispersive inequality
(1.12) keit∆fkL∞(R3) .|t|−3/2kfkL1(R3).
x x
We also record Duhamel’s formula
t
(1.13) u(t)=ei(t−t0)∆u(t )−i ei(t−s)∆(iu +∆u)(s) ds
0 t
Zt0
for any Schwartzu andany times t ,t∈R, with the conventionthat t =− t0 if
0 t0 t
t<t .
0
R R
We use the notationØ(X)to denote an expressionwhichis schematically ofthe
form X; this means that Ø(X) is a finite linear combination of expressions which
look like X but with some factors possibly replaced by their complex conjugates,
8 J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO
thusforinstance3u2v2|v|2+9|u|2|v|4+3u2v2|v|2qualifiestobeoftheformØ(u2v4),
and similarly we have
5
(1.14) |u+v|6 =|u|6+|v|6+ Ø(ujv6−j)
j=1
X
and
4
(1.15) |u+v|4(u+v)=|u|4u+|v|4v+ Ø(ujv5−j).
j=1
X
We willsometimes denote partialderivativesusing subscripts: ∂ u=∂ u=u .
xj j j
We will also implicitly use the summation conventionwhen indices are repeated in
expressions below.
We shall need the following Littlewood-Paley projection operators. Let ϕ(ξ) be
a bump function adapted to the ball {ξ ∈ R3 :|ξ|≤ 2} which equals 1 on the ball
{ξ ∈ R3 : |ξ| ≤ 1}. Define a dyadic number to be any number N ∈ 2Z of the form
N =2j wherej ∈Zisaninteger. ForeachdyadicnumberN,wedefinetheFourier
multipliers
P\f(ξ):=ϕ(ξ/N)fˆ(ξ)
≤N
P\f(ξ):=(1−ϕ(ξ/N))fˆ(ξ)
>N
P f(ξ):=(ϕ(ξ/N)−ϕ(2ξ/N))fˆ(ξ).
N
We similarly define P and P . Note in particular the telescoping identities
<N ≥N
d
P f = P f; P f = P f; f = P f
≤N M >N M M
M≤N M>N M
X X X
for all Schwartz f, where M ranges over dyadic numbers. We also define
P :=P −P = P
M<·≤N ≤N ≤M N′
M<N′≤N
X
whenever M ≤N are dyadic numbers. Similarly define P , etc.
M≤·≤N
The symbolu shallalwaysrefertoa solutionto the nonlinearSchr¨odingerequa-
tion (1.1). We shall use u to denote the frequency piece u := P u of u, and
N N N
similarly define u =P u, etc. While this may cause some confusion with the
≥N ≥N
notationu usedtodenote derivativesofu,the meaningofthe subscriptshouldbe
j
clear from context.
The Littlewood-Paley operators commute with derivative operators (including
|∇|s andi∂ +∆),thepropagatoreit∆,andconjugationoperations,areself-adjoint,
t
and are bounded on every Lebesgue space Lp and Sobolev space H˙s (if 1 ≤ p ≤
∞, of course). Furthermore, they obey the following easily verified Sobolev (and
Bernstein) estimates for R3 with s≥0 and 1≤p≤q ≤∞:
(1.16) kP fk .N−sk|∇|sP fk
≥N Lp ≥N Lp
(1.17) kP |∇|sfk .NskP fk
≤N Lp ≤N Lp
(1.18) kP |∇|±sfk ∼N±skP fk
N Lp N Lp
(1.19) kP≤NfkLq .Np3−3qkP≤NfkLp
(1.20) kPNfkLq .Np3−3qkPNfkLp.
SCATTERING FOR 3D CRITICAL NLS 9
2. Local conservation laws
In this section we record some standard facts about the (non)conservation of
mass,momentum and energy densities for generalnonlinear Schr¨odingerequations
of the form6
(2.1) i∂ φ+∆φ=N
t
on the spacetime slab I ×Rd with I a compact interval. Our primary interest is
0 0
of course the quintic defocusing case (1.1) on I ×R3 when N =|φ|4φ, but we will
0
also discuss here the U(1)-gauge invariant Hamiltonian case, when N = F′(|φ|2)φ
withR-valuedF. Lateronwewillconsidervarioustruncatedversionsof (1.1)with
non-Hamiltonianforcingterms. Theselocalconservationlawswillbeusednotonly
to imply the usual globalconservationof mass and energy, but also derive “almost
conservation” laws for various localized portions of mass, energy, and momentum,
where the localization is either in physical space or frequency space. The localized
momentuminequalitiesarecloselyrelatedtovirialidentities,andwillbeusedlater
to deduce an interaction Morawetz inequality which is crucial to our argument.
Toavoidtechnicalities(andtojustify allexchangesofderivativesandintegrals),
let us work purely with fields φ, N which are smooth, with all derivatives rapidly
decreasing in space; in practice, we can then extend the formulae obtained here
to more general situations by limiting arguments. We begin by introducing some
notationwhichwillbeusedtodescribethemassandmomentum(non)conservation
properties of (2.1).
Definition2.1. Given a (Schwartz) solution φ of (2.1) we definethe mass density
T (t,x):=|φ(t,x)|2,
00
the momentum density
T (t,x):=T (t,x):=2Im(φφ ),
0j j0 j
and the (linear part of the) momentum current
L (t,x)=L (t,x):=−∂ ∂ |φ(t,x)|2+4Re(φ φ ).
jk kj j k j k
Definition 2.2. Given any two (Schwartz) functions f,g :Rd →C, we define the
mass bracket
(2.2) {f,g} :=Im(fg)
m
and the momentum bracket
(2.3) {f,g} :=Re(f∇g−g∇f).
p
Thus {f,g} is a scalar valued function, while {f,g} defines a vector field on Rd.
m p
We will denote the jth component of {f,g} by {f,g}j.
p p
Withthesenotionswecannowexpressthemassandmomentum(non)conservation
laws for (2.1), which can be validated with straightforwardcomputations.
Lemma 2.3 (Local conservation of mass and momentum). If φ is a (Schwartz)
solution to (2.1) then we have the local mass conservation identity
(2.4) ∂ T +∂ T =2{N,φ}
t 00 j 0j m
6WewilluseφtodenotegeneralsolutionstoSchro¨dinger-typeequations,reservingthesymbol
uforsolutionstothequinticdefocusingnonlinearSchro¨dingerequation(1.1).
10 J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO
and the local momentum conservation identity
(2.5) ∂ T +∂ L =2{N,φ}j.
t 0j k kj p
Here we adopt the usual7 summation conventions for the indices j,k.
Observe that the mass current coincides with the momentum density in (2.5),
whilethemomentumcurrentin(2.5)hassome“positivedefinite”tendencies(think
of∆=∂ ∂ asanegativedefinite operator,whereasthe∂ willeventuallybe dealt
k k j
withbyintegrationby parts,reversingthe sign). Thesetwofactswillunderpinthe
interaction Morawetz estimate obtained in Section 10.
WenowspecializetothegaugeinvariantHamiltoniancase,whenN =F′(|φ|2)φ;
note that (1.1) would correspond to the case F(|φ|2)= 1|φ|6. Observe that
3
(2.6) {F′(|φ|2)φ,φ} =0
m
and
(2.7) {F′(|φ|2)φ,φ} =−∇G(|φ|2)
p
where G(z):=zF′(z)−F(z). In particular, for the quintic case (1.1) we have
2
(2.8) {|φ|4φ,φ} =− ∇|φ|6.
p
3
Thus, in the gauge invariant case we can reexpress (2.5) as
(2.9) ∂ T +∂ T =0
t 0j k jk
where
(2.10) T :=L +2δ G(|φ|2)
jk jk jk
is the (linear and nonlinear) momentum current. Integrating (2.4) and (2.9) in
space we see that the total mass
T dx= |φ(t,x)|2 dx
00
ZRd ZRd
and the total momentum
T dx=2 Im(φ(t,x)∂ φ(t,x)) dx
0j j
ZRd ZRd
are both conserved quantities. In this Hamiltonian setting one can also verify the
local energy conservation law
1 1
(2.11) ∂ |∇φ|2+ F(|φ|2) +∂ Im(φ φ )−F′(|φ|2)Im(φφ ) =0
t 2 2 j k kj j
(cid:20) (cid:21)
which implies conservation of total energy(cid:2) (cid:3)
1 1
|∇φ|2+ F(|φ|2) dx.
ZRd 2 2
Note also that (2.10) continues the tendency of the right-hand side of (2.5) to be
“positivedefinite”;this isa manifestationofthe defocusing natureofthe equation.
Later in our argument, however,we will be forced to deal with frequency-localized
versionsofthe nonlinearSchr¨odingerequations,inwhichonedoesnothaveperfect
conservationofmassandmomentum,leadingtoanumberofunpleasanterrorterms
in our analysis.
7Repeated Euclideancoordinate indicesaresummed. AsthemetricisEuclidean, wewillnot
systematicallymatchsubscriptsandsuperscripts.