Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity

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Global well-posedness for a nonlocal Gross-Pitaevskii

equation with non-zero condition at infinity

André de Laire

To cite this version:

André de Laire. Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition

at infinity. Communications in Partial Differential Equations, Taylor & Francis, 2010, 35 (11),

pp.2021-2058. �10.1080/03605302.2010.497200�. �hal-00593629�

equation with non-zero ondition at innity

André de Laire

UPMCUniv Paris 06,UMR 7598

LaboratoireJa ques-Louis Lions, F-75005, Paris, Fran e delaireann.jussieu.fr

Abstra t

We study theGross-Pitaevskii equation involvinga nonlo alintera tionpotential. Our aimis togivesu ient onditionsthat overavarietyofnonlo alintera tionssu hthattheasso iated Cau hy problemis globally well-posedwith non-zero boundary onditionatinnity, inany di-mension. Wefo usonevenpotentialsthatarepositivedeniteorpositivetempereddistributions. KeywordsNonlo al S hrödingerequation; Gross-Pitaevskiiequation; Globalwell-posedness; Initialvalueproblem.

Mathemati s Subje t Classi ation35Q55;35A05;37K05;35Q40;81Q99.

1 Introdu tion

1.1 The problem

Inorder to des ribethe kineti of aweakly intera ting Bosegas of bosons of mass

m

, Gross [22℄ and Pitaevskii [33℄ derivedin the Hartreeapproximation, that the wavefun tion

Ψ

governingthe ondensatesatises

i~∂

t

Ψ(x, t) = −

~

2

2m

∆Ψ(x, t) + Ψ(x, t)

Z

R

N

|Ψ(y, t)|

2

_{V (x − y) dy,}

on

R

N

_{× R,}

(1) where

N

is the spa e dimension and

V

des ribes the intera tion between bosons. In the most typi alrstapproximation,

V

is onsideredasaDira deltafun tion,whi hleadsto thestandard lo alGross-Pitaevskiiequation. Thislo al modelwithnon-vanishing onditionatinnityhasbeen intensivelyused, due to itsappli ation in variousareasof physi s,su h assuperuidity,nonlinear opti sandBose-Einstein ondensation[26,25,28,11℄. Itseemsthennaturaltoanalyzetheequation (1)formoregeneralintera tions. Indeed,inthestudyofsuperuidity,supersolidsandBose-Einstein ondensation,dierenttypesofnonlo alpotentialshavebeenproposed[4,13,36,34,27,1,38,12,9℄. Toobtainadimensionlessequation,wetaketheaverageenergylevelperunitmass

E

0

ofaboson, andweset

ψ(x, t) = exp



imE

0

t

~



Ψ(x, t).

Then(1)turnsinto

i~∂

t

ψ(x, t) = −

~

2

2m

∆ψ(x, t) − mE

0

ψ(x, t) + ψ(x, t)

Z

R

N

|ψ(y, t)|

2

V (x − y) dy.

(2)

u(x, t) =

1

λ

√

mE

0



_~

√

2m

2

_E

0



N

2

ψ



_~x

√

2m

2

_E

0

,

~t

mE

0



,

from(2)wededu ethat

i∂

t

u(x, t) + ∆u(x, t) + u(x, t)



1 − λ

2

Z

R

N

|u(y, t)|

2

V(x − y) dy



= 0,

with

V(x) = V



_~x

√

2m

2

_E

0



.

Ifweassumethatthe onvolutionbetween

V

anda onstantiswell-denedandequaltoapositive onstant, hoosing

λ

2

_{= (V ∗ 1)}

−1

_,

equation(2)isequivalentto

i∂

t

u + ∆u + λ

2

u(V ∗ (1 − |u|

2

)) = 0

on

R

N

× R.

(3)

More generally, we onsider the Cau hy problem for the nonlo al Gross-Pitaevskii equation with non-zeroinitial onditionatinnityintheform

(

i∂

t

u + ∆u + u(W ∗ (1 − |u|

2

)) = 0

on

R

N

× R,

u(0) = u

0

,

(NGP) where

|u

0

(x)| → 1,

as

|x| → ∞.

(4)

If

W

isareal-valuedevendistribution,(NGP)isaHamiltonianequationwhoseenergygivenby

E(u(t)) =

1

2

Z

R

N

|∇u(t)|

2

_{dx +}

1

4

Z

R

N

(W ∗ (1 − |u(t)|

2

))(1 − |u(t)|

2

) dx

isformally onserved.

Inthe asethat

W

istheDira delta fun tion,(NGP) orrespondstothelo alGross-Pitaevskii equationandtheCau hyprobleminthisinstan ehasbeenstudiedbyBéthuelandSaut[8℄,Gérard [19℄, Gallo[17℄, amongothers. As mentionedbefore, in amoregeneralframework theintera tion kernel

W

ouldbenonlo al. Forexample,Sh hesnovi handKraenkelin [36℄ onsiderfor

ε > 0

,

W

ε

(x) =















1

2πε

2

_|x|

K

0



|x|

ε



,

N = 2,

1

4πε

2

_|x|

exp



−

|x|

_ε



,

N = 3,

(5)

where

K

0

is themodiedBesselfun tion of se ondkind(also alledMa donaldfun tion). Inthis way

W

ε

might be onsidered asan approximation of the Dira delta fun tion, sin e

W

ε

→ δ

, as

ε → 0,

in adistributional sense. Othersinterestingnonlo al intera tionsarethesoft orepotential

W (x) =

(

1,

if

|x| < a,

0,

otherwise

,

(6)

with

a > 0

,whi hisusedin [27,1℄to thestudy ofsupersolids,andalso

W = α

1

δ + α

2

K,

α

1

, α

2

∈ R,

(7) where

K

isthesingularkernel

K(x) =

x

2

1

+ x

2

2

− 2x

2

3

|x|

5

,

x ∈ R

3

\{0}.

(8)

Inorder to in lude intera tions su h as(7)-(8), it is appropriateto workin thespa e

M

p,q

(R

N

₎

, that is theset of tempered distributions

W

su h that thelinear operator

f 7→ W ∗ f

is bounded from

L

p

_(R

N

₎

to

L

q

_(R

N

₎

. Wedenoteby

kW k

p,q

itsnorm. Wewillsupposethat thereexist

p

1

, p

2

, p

3

, p

4

, q

1

, q

2

, q

3

, q

4

, s

1

, s

2

∈ [1, ∞),

with

N

N − 2

> p

4

,

2N

N − 2

> p

2

, p

3

, s

1

, s

2

≥ 2, 2 ≥ q

1

>

2N

N + 2

, q

3

, q

4

>

N

2

if

N ≥ 3

and

p

2

, p

3

, s

1

, s

2

≥ 2, 2 ≥ q

1

> 1

if

2 ≥ N ≥ 1,

su h that



















W ∈ M

2,2

(R

N

) ∩

4

\

i=1

M

p

i

,q

i

(R

N

_),

1

p

3

+

1

q

2

=

1

q

1

,

1

p

1

−

1

p

3

=

1

s

1

,

1

q

1

−

1

q

3

=

1

s

2

if

N ≥ 3.

(

W

N

)

Were allthatif

p > q

, then

M

p,q

= {0}

. Thereforeifwesupposethat

W

isnotzero,thenumbers abovehavetosatisfy

q

2

, q

3

≥ 2

. Inaddition,theexisten eof

s

1

,

s

2

andtherelationsin(

W

N

)imply that

N

N − 2

> p

1

, q

2

>

N

2

,

1

p

1

−

1

p

3

∈



N − 2

2N

,

1

2



,

1

q

1

−

1

q

3

∈



N − 2

2N

,

1

2



if

N ≥ 3.

Figure1s hemati ally showsthelo ationofthesenumbersintheunitsquare.

Figure 1: For

N > 4

, the pi ture on the left represents the

(1/p, 1/q)

-plane, in the sense that

(1/p

1

, 1/q

1

) ∈ R

1

,

(1/p

2

, 1/q

2

), (1/p

3

, 1/q

3

) ∈ R

2

,

(1/p

4

, 1/q

4

) ∈ R

3

. Inthepi tureonthe right,the shadedareassymbolizethat

(1/q

1

, 1/q

3

) ∈ R

4

and

(1/p

1

, 1/p

3

) ∈ R

5

,for

N > 6

.

To he kthehypothesis (

W

N

)itis onvenientto usesomepropertiesof thespa es

M

p,q

(R

N

₎

. Forinstan e,forany

1 < p ≤ q < ∞

,

M

p,q

(R

N

_{) = M}

q

′

,p

′

(R

N

)

andforany

1 ≤ p ≤ 2

,

M

1,1

(R

N

_{) ⊆}

Asremarkedbefore,theenergyisformally onservedif

W

isareal-valuedevendistribution. We re allthatareal-valueddistributionissaidtobeevenif

hW, φi = hW, e

φi,

∀φ ∈ C

0

∞

(R

N

; R),

where

φ(x) = φ(−x)

e

. However,the onservationof energyis notsu ientto study thelongtime behavior of theCau hy problem, be ausethe potential energyis not ne essarilynonnegativeand thenonlo alnatureoftheproblempreventsustoobtainpointwisebounds. Weareableto ontrol thistermassumingfurtherthat

W

isapositivedistributionorsupposingthatitisapositivedenite distribution. Morepre isely,wesaythat

W

isapositivedistributionif

hW, φi ≥ 0,

∀φ ≥ 0, φ ∈ C

0

∞

(R

N

; R),

andthatitisapositivedenitedistributionif

hW, φ ∗ e

φi ≥ 0,

φ ∈ C

0

∞

(R

N

; R).

(9) Thesetypeof distributions frequentlyarise in thephysi al models(see Subse tion1.3). In parti -ular,thereal-valuedevenpositivedenitedistributionsin lude alargevarietyofmodelswherethe intera tionbetweenparti les issymmetri . InSe tion 2westatefurther propertiesof thiskindof potentials.

AsGallo in [17℄,we onsider theinitialdata

u

0

forthe problem(NGP) belongingto thespa e

φ + H

1

_(R

N

₎

,with

φ

afun tion ofniteenergy. Morepre isely,from nowonweassumethat

φ

isa omplex-valuedfun tion thatsatises

φ ∈ W

1,∞

(R

N

), ∇φ ∈ H

2

(R

N

) ∩ C(B

c

), |φ|

2

− 1 ∈ L

2

(R

N

),

(10) where

B

c

denotesthe omplementofsomeball

B ⊆ R

N

,sothat inparti ular

φ

satises(4). Remark1.1. Wedonotsupposethat

φ

hasalimitatinnity. Indimensions

N = 1, 2

afun tion satisfying(10) ouldhave ompli atedos illations,su has(see[19,18℄)

φ(x) = exp(i(ln(2 + |x|))

1

4

),

x ∈ R

2

.

Wenotethatanyfun tionverifying(10)belongstotheHomogeneousSobolevspa e

˙

H

1

_(R

N

_{) = {ψ ∈ L}

2

loc

(R

N

) : ∇ψ ∈ L

2

(R

N

)}.

Inparti ular, if

N ≥ 3

there exists

z

0

∈ C

with

|z

0

| = 1

su h that

φ − z

0

∈ L

2N

N −2

(R

N

)

(see e.g. Theorem4.5.9in[24℄). Choosing

α ∈ R

su hthat

z

0

= e

iα

andsin etheequation(NGP)isinvariant byaphase hange,one anassumethat

φ − 1 ∈ L

2N

N −2

(R

N

)

,butwedonotuseexpli itlythisde ay inorderto handleatthesametimethetwo-dimensional ase.

Ourmainresult on erningtheglobalwell-posednessfortheCau hyproblemis thefollowing. Theorem1.2. Let

W

bea real-valuedeven distributionsatisfying (

W

N

).

(i)

Assumethatone ofthe followingisveried

(a) N ≥ 2

and

W

isapositive denitedistribution.

(b) N ≥ 1

,

W ∈ M

1,1

(R

N

₎

and

W

isapositive distribution. Then the Cau hy problem (NGP) is globally well-posed in

φ + H

1

_(R

N

₎

. More pre isely, for every

w

0

∈ H

1

_(R

N

₎

thereexistsaunique

w ∈ C(R, H

1

_(R

N

₎₎

,for whi h

φ + w

solves (NGP) with the initial ondition

u

0

= φ + w

0

and for any bounded losed interval

I ⊂ R

, the ow map

w

0

∈ H

1

_(R

N

_{) 7→ w ∈ C(I, H}

1

_(R

N

₎₎

is ontinuous. Furthermore,

w ∈ C

1

_{(R, H}

−1

_(R

N

₎₎

andthe energyis onserved

(ii)

Assumethatthere exists

σ > 0

su hthat

ess inf c

W ≥ σ.

(12)

Then (NGP)isglobally well-posedin

φ + H

1

_(R

N

₎

,forall

N ≥ 1

and (11)holds. Moreover,if

u

isthe solutionasso iatedtothe initial data

u

0

∈ φ + H

1

_(R

N

₎

,wehavethe growth estimate

ku(t) − φk

L

2

≤ C|t| + ku

₀

− φk

_L

2

,

(13)

for any

t ∈ R

,where

C

isapositive onstant thatdepends onlyon

E

0

,

W, φ

and

σ

. WemakenowsomeremarksaboutTheorem1.2.

•

The ondition(

W

N

) impliesthat

W ∈ M

2,2

(R

N

₎

, so that

c

W ∈ L

∞

_(R

N

₎

and thereforethe ondition(12)makessense.

•

In ontrastwith(13),asweproveinSe tion 5,thegrowthestimateforthesolutiongivenby Theorem1.2-(i)isonlyexponential

ku(t) − φk

L

2

≤ C

₁

e

C

2

|t|

(1 + ku

₀

− φk

L

2

),

t ∈ R,

forsome onstants

C

1

, C

2

onlydependingon

E

0

,

W

and

φ

.

•

A ordinglytoRemark1.1andtheSobolevembeddingtheorem,afteraphase hange indepen-dentof

t

,thesolution

u

of (NGP)givenbyTheorem1.2alsosatisesthat

u − 1 ∈ L

2

N

N −2

(R

N

)

if

N ≥ 3

.

•

Indimensions

1 ≤ N ≤ 3

we an hoose

(p

4

, q

4

) = (2, 2)

in(

W

N

). Consequently,the ondition that

W ∈ M

p

4

,q

4(R

N

₎

isnontrivialonlywhen

N ≥ 4

.

Atrstsight,itisnotobviousto he kthehypotheseson

W

. Thepurposeofthenextresultis togivesu ient onditionstoensure(

W

N

).

Proposition 1.3.

(i)

Let

1 ≤ N ≤ 3

. If

W ∈ M

2,2

(R

N

_{) ∩ M}

3,3

(R

N

)

, then

W

fulls (

W

N

) . Furthermore, if

W

veries (

W

N

)with

p

i

= q

i

,

1 ≤ i ≤ 3

,then

W ∈ M

2,2

(R

N

_{) ∩ M}

3,3

(R

N

)

.

(ii)

Let

N ≥ 4

. Assumethat

W ∈ M

r,r

(R

N

₎

for every

1 < r < ∞

. Also supposethatthereexists

¯

r >

N

4

su h that

W ∈ M

p,q

(R

N

₎

, for every

1 −

1

¯

r

<

1

p

< 1

with

1

q

=

1

p

+

1

¯

r

− 1

. Then

W

satises (

W

N

).

We on lude from Proposition 1.3 that the Dira delta fun tion veries (

W

N

) in dimensions

1 ≤ N ≤ 3

. Sin e

bδ = 1

, Theorem 1.2-(ii) re overs the results of global existen e for the lo al Gross-Pitaevskiiequation in [8, 19, 17℄ and thegrowthestimate provedin [2℄. In addition, ifthe potential onvergestotheDira deltafun tion,the orrespondentsolutions onvergetothesolution ofthelo alproblemasa onsequen eofthefollowingresult.

Proposition 1.4. Assumethat

1 ≤ N ≤ 3

. Let

(W

n

)

n∈N

beasequen eofreal-valueddistributions in

M

2,2

(R

N

_{) ∩ M}

3,3

(R

N

)

su hthat

u

n

istheglobal solutionof (NGP) given byTheorem1.2, with

W

n

insteadof

W,

for someinitial datain

φ + H

1

_(R

N

₎

,and

lim

n→∞

W

n

= W

∞

,

in

M

2,2

(R

N

) ∩ M

3,3

(R

N

),

(14) with

kW

∞

k

M

₂

,2

∩M

3

,3

> 0

(

k·k

M

₂

,2

∩M

3

,3

:= max{k·k

M

2

,2, k·k

M

3

,3}

). Then

u

n

→ u

in

C(I, H

1

_(R

N

₎₎

, for any bounded losed interval

I ⊂ R

, where

u

is the solution of (NGP) with

W = W

∞

and the sameinitial data.

On the other hand, the Dira delta fun tion does not satisfy (

W

N

) if

N ≥ 4

and therefore Theorem 1.2 annot be applied. In fa t, to our knowledge there is no proof for the global well-posednesstothelo alGross-Pitaevskiiequationindimension

N ≥ 4

witharbitraryinitial ondition. Forsmall initial data, Gustafson et al. [23℄ provedglobal well-posedness in dimensions

N ≥ 4

as wellasGérard[19℄in thefour-dimensionalenergyspa e.

As a onsequen e of Theorem 1.2 and Proposition 1.3 wederivethe nextresult for integrable kernels.

Corollary 1.5. Let

W

be a real-valued even fun tion su h that

W ∈ L

1

_(R

N

₎

if

1 ≤ N ≤ 3

and

W ∈ L

1

_(R

N

_{) ∩ L}

r

_(R

N

₎

, for some

r >

N

4

, if

N ≥ 4

. Assume also that

W

is positive denite if

N ≥ 2

, or that it is nonnegative. Then the Cau hy problem (NGP) is globally well-posed in

φ + H

1

(R

N

)

.

As Galloremarks in [17℄, thewell-posednessin aspa e su h as

φ + H

1

_(R

N

₎

makespossibleto handletheproblemwithinitialdatain theenergyspa e

E(R

N

) = {u ∈ H

lo

1

(R

N

) : ∇u ∈ L

2

(R

N

), 1 − |u|

2

∈ L

2

(R

N

)},

equippedwiththedistan e

d(u, v) = ku − vk

X

1

+H

1

+ k|u|

2

− |v|

2

k

L

2

.

(15)

Here

X

1

_(R

N

₎

denotestheZhidkovspa e

X

1

(R

N

) = {u ∈ L

∞

(R

N

) : ∇u ∈ L

2

(R

N

)}.

Were allthat

u ∈ C(R, E(R

N

₎₎

is alledamildsolutionof (NGP)ifitsatisestheDuhamelformula

u(t) = e

it∆

u

0

+ i

Z

t

0

e

i(t−s)∆

(u(s)(W ∗ (1 − |u(s)|

2

)) ds,

t ∈ R.

WenotethatbyLemma6.3theintegralinther.h.sisa tuallynite(see[19,18℄forfurtherresults aboutthea tion ofS hrödingersemigroupon

E(R

N

₎

). With thesameargumentsof[17℄,wemay alsohandle the problem with initial datain theenergy spa e. Moreover,in the ase

1 ≤ N ≤ 4

, weprovethat asolutionin theenergyspa ewithinitial ondition

u

0

∈ E(R

N

₎

,ne essarilybelongs to

u

0

+ H

1

_(R

N

₎

,whi h isapropersubset of

E(R

N

₎

. This alsogivesthe uniquenessin theenergy spa efor

1 ≤ N ≤ 4

,asfollows.

Theorem 1.6. Let

W

be as in Theorem 1.2. Then for any

u

0

∈ E(R

N

₎

, there exists a unique

w ∈ C(R, H

1

_(R

N

₎₎

su h that

u := u

0

+ w

solves (NGP) . Furthermore, if

1 ≤ N ≤ 4

and

v ∈

C(R, E(R

N

₎₎

isamildsolution of (NGP) with

v(0) = u

0

,then

v = u

.

Thenextpropositionshowsthatthe hypothesesmade onthepotential

W

alsoensure the

H

2

-regularityofthesolutions.

Proposition 1.7. Let

W

be as in Theorem 1.2 and

u

be the global solution of (NGP) for some initial data

u

0

∈ φ + H

2

_(R

N

₎

. Then

u − φ ∈ C(R, H

2

_(R

N

_{)) ∩ C}

1

_{(R, L}

2

_(R

N

₎₎

.

Finally,westudythe onservationofmomentumandmassfor(NGP) . Ashasbeendis ussedin severalworks(see[5,7, 32,6℄)the lassi al on eptsofmomentumandmass,that is

p(u) =

Z

R

N

hhi∇u, uii dx

and

M (u) =

Z

R

N

(1 − |u|

2

_{) dx,}

with

hhz

1

, z

2

ii = Re(z

1

z

2

)

,arenotwell-denedfor

u ∈ φ+H

1

_(R

N

₎

. Thusitisne essarytogivesome generalizedsense to these quantities. InSe tion 7wewill explainin detailanotionof generalized momentumandgeneralizedmasssu hthatwehavethenextresultson onservationlaws.

Theorem1.8. Let

N ≥ 1

and

u

0

∈ φ + H

1

_(R

N

₎

. Then the generalizedmomentumis onservedby the owof the asso iated solution

u

of (NGP) given byTheorem1.2.

Theorem 1.9. Let

1 ≤ N ≤ 4

. In addition to (10), assume that

∇φ ∈ L

N

N −1

(R

N

)

if

N = 3, 4

. Suppose that

u

0

∈ φ + H

1

_(R

N

₎

has nite generalized mass. Then the generalized mass of the asso iated solutionof (NGP) given byTheorem1.2 is onservedby theow.

1.3 Examples

(i) Giventhespheri allysymmetri intera tionofbosons,itisusualtosupposethat

W

isradial, thatis

W (x−y) = R(|x−y|),

with

R : [0, ∞) → R

. Usingthefa tthattheFouriertransformof aradialfun tion isalsoradial,wemaywrite

W (ξ) = ρ(|ξ|),

c

forsomefun tion

ρ : [0, ∞) → R

. Noti ingthat

bδ = 1

,anextorder ofapproximationwouldbeto onsider (seee.g.[36℄)

ρ(r) =

1

1 + ε

2

_r

2

,

ε > 0.

ThentheFourierinversiontheoremimpliesthat

W

isgivenby (5)for

N = 2, 3

. By Proposi-tion2.2,(5)isindeedapositivedenitefun tion,sin e

ρ

isnonnegative. Forthispotentialwe alsohavethat

K

0

(x) ≈ ln

2

x

as

x → 0

, and

K

0

(x) ≈

p

π

2x

exp(−x)

as

x → ∞

(seee.g.[31℄, p. 136),hen e

W ∈ L

1

_(R

N

₎

for

N = 2, 3

. ThereforeitispossibletoinvokeCorollary1.5. (ii) ByLemma2.3, thefun tion givenby(6) annot bepositivedenite, sin eitisboundedand

itdoesnot oin idewithany ontinuousfun tiona.e. However,

W

isanonnegativefun tion thatbelongsto

L

1

_(R

N

_{) ∩ L}

∞

_(R

N

₎

. ThereforeCorollary1.5 anbeappliedinanydimension. (iii) Were allthatif

Ω

isanevenfun tion,smoothawayfrom theorigin,homogeneousofdegree

zero,withzeromean-valueonthesphere

Z

S

N −1

Ω(σ) dσ = 0,

then

K(x) =

Ω(x)

|x|

N

,

x ∈ R

N

\{0},

denesatempereddistribution

K

in thesenseofprin ipalvalue,that oin ideswith

K

away fromtheorigin. Moreover,forany

f ∈ S(R

N

₎

,

x ∈ R

N

,

(K ∗ f)(x) =

p.v.

Z

R

N

K(y)f (x − y) dy = lim

ε→0

Z

1

ε

>|y|>ε

Ω(y)

|y|

N

f (x − y) dy,

(16)

K ∈ M

p,p

(R

N

)

forevery

1 < p < ∞

,andtheFouriertransformof

K

belongsto

L

∞

_(R

N

₎

( f. [37℄). Therefore

W = α

1

δ + α

2

K

(17) isapositivedenitedistributionif

α

1

islargeenoughandthenTheorem1.2-(ii)givesaglobal solution of (NGP) in anydimension. Forinstan e, wemay onsider in dimensionthree the fun tion

K

givenby(8). Sin e(see[9℄)

b

K(ξ) =

4π

₃



_3ξ

2

3

|ξ|

2

− 1



,

ξ ∈ R

3

\{0},

(17)ispositivedenitebyProposition2.2if

α

1

≥

4π

3

α

2

≥ 0

or

α

1

≥ −

8π

3

α

2

≥ 0.

(18) Therefore, if (18)isveriedwemayapply Theorem 1.2-(i)-(a). Moreover,iftheinequalities in (18)arestri t,wehavealsothegrowthestimateofTheorem1.2-(ii).

needthe onstant

V ∗ 1

bepositive. Ifwetake

V

asthepotentialgivenin theexamples(i)or (ii),then

V ∈ L

1

_(R

N

₎

and

V ∗ 1 =

Z

R

N

V (x) dx > 0.

ThereforeTheorem1.2alsoprovidestheglobalwell-posednessfortheequation(1). Ifwewant to onsider

V

asintheexample(iii),themeaningof

K ∗ 1

is notobvious. However,(16)still makessenseif

f ≡ 1

. Infa t,using (16),

(K ∗ 1)(x) = lim

ε→0

Z

ε

−1

ε

Z

S

2

Ω(σ)

r

3

r

2

_{dσ dr = 0.}

Then if

V

is givenby(17),

V ∗ 1 = α

1

andwehavethesame on lusionasbefore, provided that

α

1

> 0

.

Oneoftherstworksthat introdu esthenonlo alintera tionin theGross-Pitaevskiiequation wasmade by Pomeau and Ri a in [34℄ onsidering the potential (6). Their main purpose was to establishamodelforsuperuidswithrotons. Infa t,theLandautheoryofsuperuidityofHelium II saysthat thedispersion urvemustexhibit aroton minimum(see[30, 16℄)aswas orroborated laterbyexperimentalobservations([14℄). Althoughthemodel onsideredin[34℄hasagoodtwith therotonminimum,itdoesnotprovidea orre tsoundspeed. ForthisreasonBerloin[3℄proposes thepotential

W (x) = (α + βA

2

|x|

2

+ γA

4

|x|

4

) exp(−A

2

|x|

2

),

x ∈ R

3

,

(19) where the parameters

A

,

α

,

β

and

γ

are hosen su h that the above requirements are satised. However,theexisten e of thisroton minimum impliesthat

W

c

must be negative in someinterval. Inaddition,anumeri alsimulationin[3℄showsthatin this asethesolutionexhibits nonphysi al mass on entrationphenomenon, for ertain initial onditionsin

φ + H

1

_(R

3

₎

. At somepoint,our resultsarein agreementwith theseobservationsin the sense that Theorem1.2 annot be applied tothepotential(19),be ause

c

W

and

W

arenegativeinsomeinterval. However,byProposition1.3 wemayusethefollowinglo alwell-posednessresult

Theorem 1.10. Let

W

be a distribution satisfying (

W

N

) . Then the Cau hy problem (NGP) is lo ally well-posed in

φ + H

1

_(R

N

₎

. More pre isely, for every

w

0

∈ H

1

_(R

N

₎

there exists

T > 0

su h that there is a unique

w ∈ C([−T, T ], H

1

_(R

N

₎₎

, for whi h

φ + w

solves (NGP) with the initial ondition

u

0

= φ + w

0

.

In addition,

w

is dened ona maximal timeinterval

(−T

min

, T

max

)

where

w ∈ C

1

_((−T

min

, T

max

), H

−1

(R

N

))

andthe blow-up alternativeholds:

kw(t)k

H

1

(R

N

)

→ ∞

,as

t → T

max

if

T

max

< ∞

and

kw(t)k

H

1

(R

N

)

→ ∞

,as

t → T

min

if

T

min

< ∞

. Furthermore, supposing that

W

is a real-valued even distribution, for any bounded losed interval

I ⊂ (−T

min

, T

max

)

the ow map

w

0

∈ H

1

_(R

N

_{) 7→ w ∈ C(I, H}

1

_(R

N

₎₎

is ontinuous and the energy and the generalized momentumare onserved on

(−T

min

, T

max

)

.

It is an open question to establish whi h are the exa t impli ations of hange of sign of the Fouriertransformofthepotentialfortheglobal existen eofthesolutionsof (NGP). Asproposed in[4℄, awaytohandlethis problemwould beto addahigher-ordernonlineartermin (1)to avoid themass on entrationphenomenon,maintainingthe orre tphonon-rotondispersion urve.

This paper is organized as follows. In the next se tion we give several results about positive denite and positive distributions. In Se tion 3 we establish some onvolution inequalities that involve the hypothesis (

W

N

) and we give the proof of Corollary 1.5. We prove the lo al well-posedness in Se tion 4and also Propositions 1.4and 1.7. Theorem 1.2 is ompleted in Se tion 5. InSe tion6webrieyre alltheargumentsthatleadtoTheorem1.6andinSe tion7westudythe onservationofmomentumandmass.

The purpose of this se tion is to re all some lassi al results for positive denite and positive distributions,inthe ontextofTheorem1.2. Wealsostatesomepropertiesthatwedonotuseinthe nextse tions,butareusefultobetterunderstandthetypeofpotentials onsideredinTheorem1.2.

L.S hwartzin[35℄denesthata( omplex-valued)distribution

T

ispositivedeniteif

hT, φ ∗ ˘

φi ≥ 0,

∀φ ∈ C

∞

0

(R

N

; C),

(20) with

φ(x) = φ(−x)

˘

. In virtue of our hypothesis on

W,

we have preferred to adopt the simpler denition(9). Therelationbetweenthese twopossibledenitionsisgivenin thefollowinglemma. Lemma2.1. Let

T

beareal-valueddistribution.

(i)

If

T

ispositive denite

(

inthe senseof (9)

)

andeven,then

T

fulls (20).

(ii)

If

T

veries (20),then

T

iseven.

Inparti ular, anevenreal-valueddistributionispositive denite

(

inthesense of (9)

)

if andonlyif itsatises (20).

Proof. Supposethat

T

ispositivedeniteinthesenseof(9). Let

φ ∈ C

∞

0

(R

N

; C)

,with

φ = φ

1

+iφ

2

,

φ

1

, φ

2

∈ C

0

∞

(R

N

; R)

. Then

hT, φ ∗ ˘

φi = hT, φ

1

∗ e

φ

1

i + hT, e

φ

2

∗ φ

2

i + ihT, e

φ

1

∗ φ

2

i − ihT, φ

1

∗ e

φ

2

i.

(21)

Sin e

W

iseven,

hT, e

φ

1

∗ φ

2

i = hT, φ

1

∗ e

φ

2

i.

Thereforethe imaginarypart in the r.h.s. of (21) is zero. The real part is positive be ause

T

is positivedenite,whi himpliesthat

T

veries(20).

Fortheproofof(ii),see[35℄.

The nextresult hara terizesthepositivedenite distributions under thehypothesesof Theo-rem1.2. Inparti ular,itgivesasimplewayto he kthepositivedenitenessintermsoftheFourier transform.

Proposition2.2. Let

W ∈ M

2,2

(R

N

₎

beanevenreal-valueddistribution. Thefollowingassertions areequivalent

(i) W

isapositive denitedistribution.

(ii) c

W ∈ L

∞

_(R

N

₎

and

c

W (ξ) ≥ 0

for almost every

ξ ∈ R

N

.

(iii)

Forevery

f ∈ L

2

_(R

N

_{; R)}

,

Z

R

N

(W ∗ f)(x)f(x) dx ≥ 0.

Proof. (i)

⇒

(ii). ByLemma2.1,wemayapplytheso- alledS hwartz-Bo hnerTheorem(see[35℄, p.276). Thenthereexistsapositivemeasure

µ ∈ S

′

_(R

N

₎

su hthat

W = µ

c

. Sin e

W ∈ M

2,2

(R

N

₎

, wehavethat

c

W ∈ L

∞

_(R

N

₎

(ii)

⇒

(iii). Sin e

W ∈ M

2,2

(R

N

₎

,

W ∗ f ∈ L

2

_(R

N

₎

. Fromthefa tthat

S(R

N

₎

isdensein

L

2

_(R

N

₎

, wealsohavethat

\

W ∗ f = c

W b

f .

Usingthat

f

isreal-valued,byParseval'stheoremwenally dedu e

Z

R

N

(W ∗ f)(x)f(x) dx = (2π)

−N

Z

R

N

c

W (ξ)| b

f (ξ)|

2

dξ ≥ 0,

wherewehaveusedthat

W ≥ 0

c

forthelastinequality.

(iii)

⇒

(i). Thisimpli ationdire tlyfollowsfromthefa tthat

C

∞

0

(R

N

; R) ⊂ L

2

(R

N

; R)

.

We remark that a positive denite distribution is notne essarily a positive distribution. For instan e,we onsidertheLaguerre-Gaussianfun tions

W

m

(x) = e

−|x|

2

m

X

k=0

(−1)

k

k!



m +

N

₂

m − k



|x|

2k

,

x ∈ R

N

, m ∈ N.

(22)

Thesefun tionsarenegativeinsomesubsetof

R

N

andsin e

W

c

m

≥ 0

(seee.g.[15℄,p.38), Proposi-tion2.2showsthattheyarepositivedenitefun tions. Wealsohavethat

W

m

∈ L

1

_(R

N

_)∩L

∞

_(R

N

₎

. ThenCorollary1.5givesglobal existen eof (NGP) forthepotential(22)in anydimension

N ≥ 2

. Inthe asethatthe onsidereddistributionisa tuallyaboundedfun tion,itspositivedeniteness givessomeregularity. Inotherdire tion,the on eptofpositivedenitenessmayberelatedtothe same on eptusedformatri es. Were allsomeoftheseresultsinthenextlemma.

Lemma2.3. Let

W

be anevenreal-valuedpositive denitedistribution.

(i)

If

W ∈ L

∞

_(R

N

₎

,thenit oin ides almost everywherewith a ontinuousfun tion.

(ii)

If

W

is ontinuous,then

W (0) = kW k

L

∞

(R

N

)

andforall

x

1

, . . . , x

m

∈ R

N

,

m ≥ 1

,thematrix given by

A

jk

= W (x

j

− x

k

)

,

j, k ∈ {1, . . . , m}

,isapositive semi-denitematrix.

Proof. Takinginto onsiderationLemma2.1, thesestatementsareprovedin[35℄.

The importan e of the ondition (12) is that it givesthe following oer ivity property to the potentialenergy.

Lemma2.4. Assumethat

W ∈ M

2,2

(R

N

₎

veries (12). Then for all

f ∈ L

2

_(R

N

_{; R)}

,

σkfk

2

L

2

≤

Z

R

N

(W ∗ f)(x)f(x) dx ≤ kW k

2,2

kfk

2

L

2

.

(23)

Proof. Therstinequalityfollowsfrom Parseval'stheorem,

Z

R

N

(W ∗ f)(x)f(x) dx = (2π)

−N

Z

R

N

c

W (ξ)| b

f (ξ)|

2

dξ ≥ σkfk

2

L

2

.

These ond inequalityin(23)isimmediatesin e

W ∈ M

2,2

(R

N

₎

.

The purpose of the last lemma in this se tion is to establish some properties of the positive distributions whi h appear in Theorem 1.2. In parti ular, we show that for these distributions (

W

N

)isautomati ally veriedif

1 ≤ N ≤ 3

.

Lemma 2.5. Let

W ∈ M

1,1

(R

N

₎

be a positive distribution. Then

W ∈ M

p,p

(R

N

₎

, for any

1 ≤

p ≤ ∞

and

W

isapositiveBorelmeasureofnitemass. If

1 ≤ N ≤ 3

wealsohavethat

W

satises (

W

N

).

Proof. Sin e

W ∈ M

1,1

(R

N

₎

, itis well knownthat

W

isa( omplex-valued)niteBorel measure. Then

W ∈ M

∞,∞

(R

N

₎

and by interpolation

W ∈ M

p,p

(R

N

₎

forany

1 ≤ p ≤ ∞

. Finally, thefa t that

W

isapositivedistributionimpliesthatitisapositivemeasure ( f. [35℄). ByProposition1.3 we on ludethat

W

satises(

W

N

) ,if

1 ≤ N ≤ 3

.

3 Some onsequen es of assumption (

W

N

)

Werstestablishsomeinequalitiesinvolvingthe onvolutionwith

W

thatexplainin parthowthe hypothesis(

W

N

)works.Afterthat,wegivetheproofofProposition1.3andCorollary1.5.

From now on weadopt the standard notation

C(·, ·, . . . )

to representa generi onstant that dependsonlyonea hofitsarguments,andpossiblyonsomexednumberssu hasthedimension. Inthe asethat

W ∈ M

p,q

(R

N

₎

weuse

C(W )

todenotea onstantthatonlydependsonthenorm

kW k

p,q

. Wealsousethenotation

p

′

forthe onjugateexponentof

p

givenby

1/p + 1/p

′

_{= 1}

. Lemma3.1. Let

W ∈ M

p

₁

,q

₁

(R

N

) ∩ M

p

2

,q

2(R

N

) ∩ M

p

3

,q

3(R

N

₎

,with

p

1

, p

2

, p

3

, q

1

, q

2

, q

3

≥ 1

and

1

p

3

+

1

q

2

=

1

q

1

.

Supposethatthereare

s

1

, s

2

≥ 1

,su hthat

1

p

1

−

1

p

3

=

1

s

1

,

1

q

1

−

1

q

3

=

1

s

2

.

Then forany

u, v ∈ S(R

N

₎

k(W ∗ u)vk

L

q1

≤ kW k

_p

2

,q

2

kuk

L

p2

kvk

L

p3

,

k(W ∗ u)vk

L

q1

≤ kW k

_p

3

,q

3

kuk

L

p3

kvk

L

s2

,

kW ∗ (uv)k

L

q1

≤ kW k

_p

1

,q

1

kuk

L

p3

kvk

L

s1

.

Proof. Theproofisadire t onsequen eofHölderinequalityand thehypotheseson

W

. Lemma 3.2. Assume that

W

satises (

W

N

) and that

N ≥ 4

. Then

W ∈ M

N

N −2

,2

(R

N

₎

,

W ∈

M

N

N −2

,

N

2

(R

N

₎

and

W ∈ M

2,

N

2

(R

N

₎

.

Proof. FromtheRiesz-Thorininterpolationtheoremandthefa tthat

1

2

,

2

N

and

N −2

N

,

2

N

belong tothe onvexhullof



1

2

,

1

2



,



₁

p

1

,

1

q

1



,



₁

p

3

,

1

q

3



,



₁

p

4

,

1

q

4



,

we on ludethat

W ∈ M

2,

N

2

(R

N

₎

and

W ∈ M

N

N −2

,

N

2

(R

N

₎

. Sin ethe onjugateexponentof

N

N −2

is

N

2

,

W ∈ M

2,

N

2

(R

N

₎

impliesthat

W ∈ M

N

N −2

,2

(R

N

₎

.

Lemma3.3. Assumethat

W

satises (

W

N

). Then for any

u, v, w ∈ S(R

N

₎

,

k(W ∗ (uv))wk

L

˜

γ

≤ C(W )kuk

_L

˜

s

kvk

_L

r

˜

kwk

_L

r

˜

,

(24) for some

2 > ˜

γ >

2N

N +2

,

2N

N −2

> ˜

r, ˜

s > 2

if

N ≥ 3

,and

2 > ˜

γ > 1

,

∞ > ˜r, ˜s > 2

if

N = 1, 2

.

Proof. If

N ≥ 4

, byLemma 3.2 we havethat

W ∈ M

N

N −2

,

N

2

(R

N

₎

. Sin e also

W ∈ M

p

4

,q

4(R

N

₎

, fromtheRiesz-Thorininterpolationtheoremwededu ethatthereexist

p

¯

and

q

¯

su hthat

W ∈ M

p,¯

¯

q

(R

N

),

N

N − 1

< ¯

p <

N

N − 2

,

N

2

< ¯

q < N.

(25) Nowweset

1

˜

r

= min



₁

2



1 −

1

_q

_¯



,

1

2¯

p



,

1

˜

γ

=

1

¯

q

+

1

¯

r

,

1

˜

s

=

1

¯

p

−

1

˜

r

.

Inviewof (25),wehave

2N

N +2

< ˜

γ < 2

and

2 < ˜

r, ˜

s <

N −2

2N

. ByHölderinequality,we on ludethat

k(W ∗ (uv))wk

L

˜

γ

≤ kW ∗ (uv)k

_L

q

¯

kwk

_L

r

˜

≤ kW k

p,¯

¯

q

kuvk

L

p

¯

kwk

_L

˜

r

≤ kW k

p,¯

¯

q

kuk

L

s

˜

kvk

_L

r

˜

kwk

_L

r

˜

.

If

N = 1, 2, 3

,theproofissimpler. Itissu ienttotake

q = 2

¯

,

p = 2

¯

,

s = ˜

˜

r = 4

,

γ =

˜

4

3

in thelast inequalitytodedu e(24).

Lemma3.4. Assumethat

W

satises (

W

N

).

(i)

Forany

u ∈ φ + H

1

_(R

N

₎

wehave

(W ∗ (1 − |u|

2

_{))(1 − |u|}

2

_{) ∈ L}

1

_(R

N

₎

(ii)

If

W

isalso anevenreal-valueddistribution, thenfor any

u ∈ φ + H

1

_(R

N

₎

and

h ∈ H

1

_(R

N

₎

,

Z

R

N

(W ∗ hhu, hii)(1 − |u|

2

_{) dx =}

Z

R

N

(W ∗ (1 − |u|

2

))hhu, hii dx.

(26)

Proof. Let

u = φ + w

,with

w ∈ H

1

_(R

N

₎

. If

N ≥ 4

, by (10)andtheSobolevembedding theorem, wededu ethat

(1 − |φ|

2

− 2hhφ, wii − |w|

2

) ∈ L

2

(R

N

) + L

N −2

N

(R

N

).

By Lemma 3.2 we have that the map

h 7→ W ∗ h

is ontinuous from

L

2

_(R

N

_{) + L}

N −2

N

(R

N

)

to

L

2

_(R

N

_{) ∩ L}

N

2

(R

N

)

andsin e

N −2

N

+

2

N

= 1

,byHölderinequalitywe on ludethat

(W ∗ (1 − |φ|

2

− 2hhφ, wii − |w|

2

))(1 − |φ|

2

− 2hhφ, wii − |w|

2

) ∈ L

1

(R

N

).

(27)

If

1 ≤ N ≤ 3

,(27)followsfromthefa tthat

|w|

2

_{∈ L}

2

_(R

N

₎

. This on ludestheproofof(i). Asimilarargumentshowsthat

k(W ∗ hhu, hii)(1 − |u|

2

_)k

L

1

< ∞

. Thenusingthat

W

isevenand Fubini's theoremweobtain(ii).

The previouslemmas will be useful in the next se tions, in parti ular to prove thelo al well-posednessof(NGP). NowwegivetheproofsofProposition1.3andCorollary1.5,thatinvolvesome straightforward omputations.

Proof ofProposition1.3. For the rst part of (i), we note that the hypothesis implies that

W ∈

M

p,p

(R

N

)

for any

3

2

≤ p ≤ 3

. Then itis su ient to take

p

1

= q

1

=

3

2

,

p

2

= p

3

= q

2

= q

3

= 3

and

p

4

= q

4

= 2

to see that (

W

N

) is fullled. For the se ond part of (i), we need prove that

W ∈ M

3,3

(R

N

)

. Re allingthat

M

p,q

(R

N

_{) = M}

q

′

_,p

′

(R

N

)

for

1 < p ≤ q < ∞

and using theRiesz interpolationtheorem,wehavethat

W ∈ M

s,t

(R

N

₎

,forevery

(s

−1

_{, t}

−1

₎

inthe onvexhullof



₁

2

,

1

2



∪

3

[

j=1



₁

p

j

,

1

q

j



,



1 −

_q

1

j

, 1 −

1

p

j



.

(28)

Byhypothesis,

p

i

= q

i

,

i = 1, 2, 3

,thus(

W

N

)impliesthat

1

p

2

+

1

p

3

=

1

p

1

,

2 ≥ p

1

and

p

2

, p

3

≥ 2.

Hen ethe onvexhullof (28)simpliesto



(x, x) ∈ R

2

: min



1 −

_p

1

1

,

1

p

2

,

1

p

1

−

1

p

2



≤ x ≤ max



₁

p

1

, 1 −

1

p

2

, 1 −

1

p

1

+

1

p

2



.

Arguingby ontradi tion,itissimpletoseethat

min



1 −

_p

1

1

,

1

p

2

,

1

p

1

−

1

p

2



≤

1

₃

and

2

3

≤ max



1

p

1

, 1 −

1

p

2

, 1 −

1

p

1

+

1

p

2



.

Therefore

W ∈ M

s,s

(R

N

₎

,forevery

3

2

≤ s ≤ 3

. Inparti ular

W ∈ M

2,2

(R

N

_{) ∩ M}

3,3

(R

N

)

. Toprove(ii),wenoti ethatbyinterpolationwehavethat

W ∈ M

α,β

(R

N

₎

,forall

α, β

satisfying

1 ≤ α, β,

1

α

−



1 −

1

¯

r



≤

1

β

≤

1

α

.

(29) Wenowdene

p

2

= p

3

=

(

3,

if

4 ≤ N ≤ 5,

s

N

s

N

−1

,

if

6 ≤ N,

q

2

= q

3

=

(

3,

if

4 ≤ N ≤ 5,

N,

if

6 ≤ N,

p

1

=

(

₃

2

,

if

4 ≤ N ≤ 5,

N

N −1

,

if

6 ≤ N,

q

1

=

(

₃

2

,

if

4 ≤ N ≤ 5,

p

3

q

2

p

3

+q

2

,

if

6 ≤ N,

p

4

=

_2¯

_r−1

2¯

r

,

q

4

= 2¯

r

,where

s

N

=











N

4

+ ε

N

,

if

6 ≤ N ≤ 7,

2(N + 1)

N + 2

,

if

8 ≤ N,

and

ε

N

> 0

is hosensmallenoughsu h that

0 < ε

N

< 2 −

N

4

if

6 ≤ N ≤ 7

. Thenwehavethat

2N

N + 2

< s

N

< 2,

forany

N ≥ 6.

(30) Usingthat

¯

r >

N

4

and(30),we anverifythatthe hoi eof

(p

i

, q

i

)

,

i ∈ {1, . . . , 4}

,satises(29)with

α = p

i

and

β = q

i

, aswellasallthe othersrestri tionsin the hypothesis (

W

N

), whi h ompletes theproof.

Proof ofCorollary 1.5. ByYounginequalitywehavethat

W ∈ M

p,p

(R

N

₎

,forany

1 ≤ p ≤ ∞

. In parti ularthe ondition

W ∈ M

1,1

(R

N

₎

isfullled. If

1 ≤ N ≤ 3

,the on lusion isa onsequen e ofProposition 1.3andTheorem 1.2. If

N ≥ 4

, byYounginequalitywehavethat

W ∈ M

p,q

(R

N

₎

, forall

1 −

1

r

≤

1

p

≤ 1

, with

1

q

=

1

p

+

1

r

− 1

. Thentheprooffollowsagainfrom Proposition 1.3and Theorem1.2.

4 Lo al existen e

Inorder to proveTheorem 1.2 we rstare goingto provethelo al well-posedness. Theorem1.10 isbasedonthefa t that ifweset

u = w + φ

,then

u

isasolutionof (NGP) with initial ondition

u

0

= φ + w

0

ifandonlyif

w

solves

(

i∂

t

w + ∆w + f (w) = 0

on

R

N

× R,

w(0) = w

0

,

(31)

f (w) = ∆φ + (w + φ)(W ∗ (1 − |φ + w|

2

_)).

Wede ompose

f

as

f (w) = g

1

(w) + g

2

(w) + g

3

(w) + g

4

(w),

(32) with

g

1

(w) = ∆φ + (W ∗ (1 − |φ|

2

))φ,

g

2

(w) = −2(W ∗ hhφ, wii)φ,

g

3

(w) = −(W ∗ |w|

2

)φ − 2(W ∗ hhφ, wii)w + (W ∗ (1 − |φ|

2

))w,

g

4

(w) = −(W ∗ |w|

2

)w.

Thenextlemmagivessomeestimatesonea hofthese fun tions.

Lemma4.1. Assumethat

W

satises (

W

N

). Usingthe numbers given by (

W

N

)and Lemma 3.3, let

r

1

= r

2

= 2

,

r

3

= p

3

,

r

4

= ˜

r

,

ρ

1

= ρ

2

= 2

,

ρ

3

= q

′

1

and

ρ

4

= ˜

γ

′

_.

Then

g

j

∈ C(H

1

(R

N

), H

−1

(R

N

)), j ∈ {1, 2, 3, 4}.

(33)

Furthermore, for any

M > 0

thereexistsa onstant

C(M, W, φ)

su hthat

kg

j

(w

1

) − g

j

(w

2

)k

L

ρ

′

j

≤ C(M, W, φ)kw

1

− w

2

k

_L

rj

,

(34) for all

w

1

, w

2

∈ H

1

_(R

N

₎

with

kw

1

k

H

1

, kw

2

k

H

1

≤ M

,and

kg

j

(w)k

W

1,ρ

′

j

≤ C(M, W, φ)(1 + kwk

_W

1

_,rj

),

(35) for all

w ∈ H

1

_(R

N

_{) ∩ W}

1,r

j

_(R

N

₎

with

kwk

H

1

≤ M

.

Proof. Sin e

g

1

isa onstant fun tion of

w

,

g

1

∈ C(H

1

_(R

N

_{), H}

−1

_(R

N

₎₎

and (34)is trivialin this ase. The ondition(35)followsfromtheestimate

kg

1

(w)k

H

1

≤k∇φk

_H

2

+ kW k

_2,2

(k1 − |φ|

2

k

_L

2

kφk

_W

1,∞

+ 2kφk

2

_L

∞

k∇φk

_L

2

).

Similarlyweobtainfor

g

2

,

kg

2

(w

1

) − g

2

(w

2

)k

L

2

≤ 2kW k

_2,2

kφk

2

L

∞

kw

₁

− w

₂

k

_L

2

and

k∇g

2

(w)k

L

2

≤ 2kW k

_2,2

kφk

_L

∞

kφk

_L

∞

k∇wk

_L

2

+ 2k∇φk

_L

∞

kwk

_L

2

≤ C(W, φ)kwk

H

1

.

Thenwededu e(34)and(35)for

j = 2

. For

g

3

,wehave

g

3

(w

2

) − g

3

(w

1

) = (W ∗ (|w

1

|

2

− |w

2

|

2

))φ + 2(W ∗ hhφ, w

1

− w

2

ii)w

1

+ 2(W ∗ hhφ, w

2

ii)(w

1

− w

2

) + (W ∗ (1 − |φ|

2

))(w

1

− w

2

).

Theassumption(

W

N

)allowstoapplyLemma3.1andthenwederive

kg

3

(w

2

) − g

3

(w

1

)k

_L

ρ

′

3

≤ C(W, φ)kw

1

− w

2

k

L

r3

(kw

1

k

L

s1

+ kw

2

k

L

s1

+2kw

1

k

L

s2

+ 2kw

₂

k

_L

p2

+ 1).

Morepre isely,thedependen eon

φ

ofthe onstant

C(W, φ)

inthelastinequalityisgivenexpli itly by

max{kφk

L

∞

, k1 − |φ|

2

_k

L

p2

}

. BytheSobolevembeddingtheorem

H

1

(R

N

) ֒→ L

p

(R

N

),

∀ p ∈



2,

2N

N − 2



if

N ≥ 3

and

∀ p ∈ [2, ∞)

if

N = 1, 2.

(37) Inparti ular,

kw

1

k

L

s1

+ kw

₂

k

_L

s1

+ 2kw

₁

k

_L

s2

+ 2kw

₂

k

_L

p2

≤ C(kw

₁

k

_H

1

+ kw

₂

k

_H

1

),

whi h together with (36) gives us (34)for

g

3

. With the same type of omputations, taking

w ∈

H

1

_(R

N

₎

,

kwk

H

1

≤ M

,wehave

k∇g

3

(w)k

_L

ρ

′

3

≤C(M, W, φ)(k∇wk

L

r3

+ kwk

L

r3

),

wherethedependen e on

φ

isin termsof

kφk

L

∞

,

k∇φk

L

∞

,

k1 − |φ|

2

_k

L

p2

and

k∇φk

L

p2

. For

g

4

,applyingLemma3.3weobtain

kg

4

(w

1

) − g

4

(w

2

)k

_L

ρ

′

4

≤ C(W )kw

1

− w

2

k

L

r4

((kw

1

k

L

s

+ kw

2

k

L

s

)kw

1

k

L

r4

+kw

2

k

L

s

kw

₂

k

_L

r4

)

and

k∇g

4

(w)k

_L

ρ

′

4

≤C(W )k∇wk

L

r4

kwk

L

r4

kwk

L

s

.

Asbefore, using(37),we on ludethat

g

4

veries(34)-(35). Sin efor

2 ≤ j ≤ 4

,

2 ≤ r

j

<

2N

N −2

(

2 ≤ r

j

< ∞

if

N = 1, 2

),wehavethe ontinuousembeddings

H

1

(R

N

) ֒→ L

r

j

_(R

N

₎

and

L

r

′

j

_(R

N

_{) ֒→ H}

−1

_(R

N

_).

Theninequality(34)implies(33),for

j ∈ {2, 3, 4}

.

Nowweanalyzethepotentialenergyasso iatedto (31). Forany

v ∈ H

1

_(R

N

₎

weset

F (v) :=

Z

R

N

hh∆φ, vii dx −

1

4

Z

R

N

(W ∗ (1 − |φ + v|

2

))(1 − |φ + v|

2

) dx,

(38)

andusingthenotationofLemma 4.1,wexfortherestofthisse tion

r = max{r

1

, r

2

, r

3

, r

4

, ρ

1

, ρ

2

, ρ

3

, ρ

4

}.

(39)

Lemma4.2. Assumethat

W

satises (

W

N

). Then the fun tional

F

is well-denedon

H

1

_(R

N

₎

. If moreover

W

isareal-valuedeven distribution,wehave the following properties.

(i) F

isFré het-dierentiableand

F ∈ C

1

(H

1

(R

N

), R)

with

F

′

_{= f.}

(40)

(ii)

Forany

M > 0

,thereexistsa onstant

C(M, W, φ)

su hthat

|F (u) − F (v)| ≤ C(M, W, φ)(ku − vk

L

2

+ ku − vk

_L

r

),

(41) for any

u, v ∈ H

1

_(R

N

₎

,with

kuk

H

1

, kvk

H

1

≤ M

.

Proof. ByLemma 3.4,

F

iswell-denedin

H

1

_(R

N

₎

forany

N

. Toprove(i), we omputenowthe Gâteauxderivativeof

F

. For

h ∈ H

1

_(R

N

₎

wehave

d

G

F (v)[h] = lim

t→0

F (v + th) − F (v)

t

=

Z

R

N

hh∆φ, hii dx +

1

2

Z

R

N

(W ∗ hhφ + v, hii)(1 − |φ + v|

2

_{) dx}

+

1

2

Z

R

N

(W ∗ (1 − |φ + v|

2

))hhφ + v, hii dx.

Sin e

W

is aneven distribution,(26)impliesthat thelast twointegralsareequal. Finally weget that

d

G

F (v)[h] =

Z

R

N

hhf(v), hii dx = hf(v), hi

H

−1

_,H

1

.

From (32) and (33), we have that

f ∈ C(H

1

_(R

N

_{), H}

−1

_(R

N

₎₎

. Hen e the map

v → d

G

F (v)

is ontinuousfrom

H

1

_(R

N

₎

to

H

−1

_(R

N

₎

, whi h impliesthat

F

is ontinuouslyFré het-dierentiable andsatises(40).

Fortheproofof(ii),using(40)andthemean-valuetheorem,wehave

F (u) − F (v) =

Z

1

0

d

ds

F (su + (1 − s)v) ds =

Z

1

0

hf(su + (1 − s)v), u − vi

H

−1

,H

1

ds.

ThenbyLemma 4.1,

|F (u) − F (v)| ≤ sup

s∈[0,1]

4

X

j=1

kg

j

(su + (1 − s)v)k

L

ρ

′

j

ku − vk

L

ρj

≤

4

X

j=1

C(M, W, φ)(kuk

L

rj

+ kvk

L

rj

+ 1)ku − vk

L

ρj

.

(42)

Sin eweassumethat

kuk

H

1

, kvk

H

1

≤ M

,(37)impliesthat

kuk

L

rj

+ kvk

L

rj

+ 1 ≤ C(M).

(43)

Also,itfollowsfrom

L

p

-interpolationandYoung'sinequalitythat

ku − vk

L

ρj

≤ ku − vk

θ

j

L

2

ku − vk

1−θ

j

L

r

≤ ku − vk

L

2

+ ku − vk

_L

r

,

(44) with

θ

j

=

2(r−ρ

j

)

ρ

j

(r−2)

.

By ombining(42),(43)and(44),weobtain(ii).

Proof ofTheorem1.10. Re alling that

r

wasxed in (39), we dene

q

by

1

q

=

N

2

1

2

−

1

r

. Given

T, M > 0

,we onsiderthe ompletemetri spa e

X

T,M

= {w ∈ L

∞

((−T, T ), H

1

(R

N

)) ∩ L

q

((−T, T ), W

1,r

(R

N

)) :

kwk

L

∞

((−T,T ),H

1

)

≤ M, kwk

L

q

_{((−T,T ),W}

1

,r

₎

≤ M},

endowedwiththedistan e

d

T

(w

1

, w

2

) = kw

1

− w

2

k

L

∞

_{((−T,T ),L}

2

₎

+ kw

₁

− w

₂

k

_L

q

_{((−T,T ),L}

r

₎

.

(45)

Theestimatesgivenin Lemmas4.1,4.2andtheStri hartzestimatesshowthatthefun tional

Φ(w) = e

it∆

w

0

+ i

Z

t

0