on collapsing ring blow-up solutions to the mass supercritical nonlinear schrödinger equation

ON COLLAPSING RING BLOW-UP SOLUTIONS TOTHE MASS SUPERCRITICAL NONLINEARSCHRÖDINGER EQUATION

FRANK MERLE, PIERRE RAPHAËL, and JEREMIE SZEFTEL

AbstractWe consider the nonlinear Schrödinger equation i@tuC�uCujujp�1 D 0 in dimen-sion N � 2 and in the mass supercritical and energy subcritical range 1C 4

N< p <

min¹NC2N�2

; 5º. For initial data u0 2 H 1 with radial symmetry, we prove a univer-sal upper bound on the blow-up speed. We then prove that this bound is sharp andattained on a family of collapsing ring blow-up solutions first formally predicted inFibich et al.

Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3692. Universal upper bound on the blow-up rate . . . . . . . . . . . . . . . . 3803. The approximate solution . . . . . . . . . . . . . . . . . . . . . . . . 3834. Setting up the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3945. Monotonicity formula . . . . . . . . . . . . . . . . . . . . . . . . . . 4036. Existence of ring solutions . . . . . . . . . . . . . . . . . . . . . . . . 418Appendix A. Integration of the exact system of modulation equations . . . . . 422Appendix B. Stability of the modulation equations . . . . . . . . . . . . . . 426References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

1. Introduction

1.1. Setting of the problemWe consider in this paper the nonlinear Schrödinger equation (NLS)

(NLS)

´i@tuC�uC juj

p�1uD 0;

ujtD0 D u0;.t; x/ 2R�RN (1.1)

DUKE MATHEMATICAL JOURNALVol. 163, No. 2, © 2014 DOI 10.1215/00127094-2430477Received 13 March 2012. Accepted 13 June 2013.2010 Mathematics Subject Classification. Primary 35Q55; Secondary 35Q51.Authors’ work supported by the European Research Council (ERC) Advanced Grant BLOWDISOL. Raphaël’s

and Szeftel’s work also supported by the ERC and Agence Nationale de la Recherche SWAP program.

369

http://dx.doi.org/10.1215/00127094-2430477

370 MERLE, RAPHAËL, and SZEFTEL

in dimension N � 2 and in the mass supercritical and energy subcritical range

1C4

N< p < 2� � 1; 2� D

´C1 for N D 2;2NN�2

for N � 3:(1.2)

From Ginibre and Velo [10], given u0 2 H 1, there exists a unique solution u 2C.Œ0; T /;H 1/ to (1.1) and there holds the blow-up alternative:

T <C1 implies limt!T

��u.t/��H1DC1:

The H 1 flow admits the conservation laws:

Mass: M.u/Z ˇ̌u.t; x/

ˇ̌2DM.u0/;

Energy: E.u/D1

2

Z ˇ̌ru.t; x/

ˇ̌2dx �

1

pC 1

Z ˇ̌u.t; x/

ˇ̌pC1dx DE.u0/;

Momentum: P.u/D=�Zru.t; x/u.t; x/dx

�D P.u0/:

A large group of symmetries also acts in the energy space H 1, in particular, the scal-ing symmetry

u.t; x/ 7! �2p�1

0 u.�20t; �0x/; �0 > 0; (1.3)

and the Galilean drift

u.t; x/ 7! u.t; x � ˇ0t /eiˇ02 �.x�

ˇ02 t/; ˇ0 2R

N :

The scaling invariant homogeneous Sobolev space PH sc attached to (1.1) is the onewhich leaves the scaling symmetry invariant, explicitly:

sc DN

2�

2

p � 1:

We say that the problem is mass subcritical if sc < 0, mass critical if sc D 0, andmass supercritical if sc > 0. From standard argument, for mass subcritical problems,the energy dominates the kinetic energy and all H 1 solutions are global and bounded(see [4]). On the other hand, for sc � 0 and data

u0 2†DH1 \ ¹xu 2L2º;

the celebrated virial identity

ON COLLAPSING RING BLOW-UP SOLUTIONS 371

d2

dt2

Zjxj2

ˇ̌u.t; x/

ˇ̌2dx D 4N.p � 1/E.u0/�

16sc

N � 2sc

Zjruj2

� 16E.u0/ (1.4)

implies that solutions emerging from nonpositive energy initial dataE.u0/ < 0 cannotexist globally and hence blow up in finite time.

This dichotomy can also be seen on the stability of ground state periodic solutionsu.t; x/ D Q.x/eit , where Q is, from [9] and [17], the unique (up to symmetries)solution to

�Q�QCQp D 0; Q 2H 1;Q > 0: (1.5)

From variational arguments [5], these solutions are orbitally stable for sc < 0, andunstable by blowup and scattering for sc > 0, [2] and [31].

Note that we may reformulate the condition (1.2) as

0 < sc < 1:

In this setting, the Cauchy problem is also well posed in PH s for sc � s � 1 and, fromstandard argument, this implies the scaling lower bound on the blow-up speed forH 1

finite time blow-up solutions:

��ru.t/��L2

� 1

.T � t /1�sc2

I (1.6)

see [27] for further details.

1.2. Qualitative information on blowupThere is still little understanding of the blow-up scenario for general initial data. Thesituation is better understood in the mass critical case sc D 0 since the series of works[22]–[26] and [32] where a stable blow-up regime of “log-log” type is exhibited indimension N � 5 with a complete description of the associated bubble of concentra-tion. In particular, blowup occurs at a point and the solution concentrates exactly theground state mass ˇ̌

u.t; x/ˇ̌2* kQk2

L2ıxDx� C ju

�j2 as t! T (1.7)

for some .x�; u�/ 2 RN � L2. This blow-up dynamic is not the only one and thereexist further threshold dynamics which transition from stable blowup to stable scat-tering (see [3], [30]). These explicit scenarios correspond to an improved descriptionof the flow near the ground state solitary wave.


For sc > 0, the situation is more poorly understood. The only general featureknown on blowup is the existence of a universal upper bound on blow-up rate,�Z T

0

.T � t /��ru.t/��2

L2dt <C1; (1.8)

which is a direct consequence of the time integration of the virial identity (1.4)(see [4]). In [27], Merle and Raphaël considered radial data in the range 0 < sc < 1,and showed that if blowup occurs, the Sobolev invariant critical norm does not con-centrate as in (1.7), it actually blows up with a universal lower bound��u.t/�� PH sc � ˇ̌log.T � t /

ˇ̌C.N;p/: (1.9)

This relates to the regularity results for the three-dimensional Navier–Stokes (see [7])and the regularity result (see [13]), and shows a major dynamical difference betweencritical and supercritical blowup. Then two explicit blow-up scenarios have been con-structed so far. In [29], a stable self-similar blow-up regime

��ru.t/��L2�

1

.T � t /1�sc2

is exhibited in the range 0 < sc � 1, N � 5, which bifurcates in some sense fromthe log-log analysis in [23] and [25]. These solutions concentrate again at a point inspace.

A completely different scenario is investigated in [35] and [37] for the quinticnonlinearity p D 5 in dimensions N � 2 where “standing ring” solutions are con-structed. These solutions have radial symmetry and concentrate their mass on anasymptotic fixed sphere

u.t; r/�1

�2p�1 .t/

Q�r � r��.t/

�; r� > 0;

where Q is the one dimensional mass critical ground state p D 5, and the speed ofconcentration is given by the log-log law

�.t/�

sT � t

log j log.t � t /j:

Note that this includes energy-critical (N D 3) and energy-supercritical regimes(N � 4), and this blow-up scenario is shown to be stable by smooth radially sym-metric perturbation of the data. We refer to [40], [11], and [12] for further extensionsin cylindrical symmetry.

�For data u0 2†DH1 \ ¹xu 2L2º.


In the breakthrough paper [8], Fibich, Gavish, and Wang proposed a formal gen-eralization of the ring scenario for 1C 4

N< p < 5: they formally predicted and numer-

ically observed solutions with radial symmetry which concentrate on a collapsing ring

u.t; r/�1

�2p�1 .t/

.Qe�ˇ1y/�r � r.t/

�.t/

�;

where Q is the mass subcritical one-dimensional ground state solution to (1.5), ˇ1is a universal Galilean drift

ˇ1 D

s5� p

pC 3; (1.10)

and concentration occurs at the speed

�.t/� .T � t /1

1C˛ ; r.t/D .T � t /˛1C˛

for some universal interpolation number

˛D5� p

.p � 1/.N � 1/: (1.11)

Moreover, numerics suggest that this blowup is stable by radial perturbation of thedata. This blowup corresponds to a new type of concentration, and like the standingring solution for pD 5, it recovers in the supercritical regime the mass concentrationscenario (1.7).

1.3. Statement of the resultWe first claim a universal space-time upper bound on the blow-up rate for radial datain the regime 0 < sc < 1 which sharpens the rough virial bound (1.8).

THEOREM 1.1 (Upper bound on blow-up rate for radial data)Let

N � 2; 0 < sc < 1; p < 5:

Let u0 2H 1 with radial symmetry, and assume that the corresponding solution u 2C.Œ0; T /;H 1/ of (1.1) blows up in finite time t D T . Then there holds the space-timeupper bound

Z T

t

.T � �/��ru.�/��2

L2d� � C.u0/.T � t /

2˛1C˛ ; (1.12)

where ˛ is given by (1.11).


The proof of (1.12) is surprisingly simple and relies on a sharp version of thelocalized virial identity introduced in [27]. Recall that no upper bound on the blow-uprate is known in the mass critical case sc D 0, and arbitrary slow type II concentration�

should be expected for the energy-critical problem sc D 1 in the continuation of [16].Note also that the bound (1.12) implies

lim inft"T

.T � t /1

1C˛

��ru.t/��L2<C1;

but the derivation of a pointwise upper bound on blow-up speed for all times remainsopen.

We now claim that the bound (1.12) is sharp in all dimensions and attained onthe collapsing ring solutions.

THEOREM 1.2 (Existence of collapsing ring blow-up solutions)Let

N � 2; 0 < sc < 1; p < 5;

and ˇ1 > 0, 0 < ˛ < 1 given by (1.10) and (1.11). Let Q be the one-dimensionalmass subcritical ground state solution to (1.5). Then there exists a time t < 0 and asolution u 2 C.Œt ; 0/;H 1/ of (1.1) with radial symmetry which blows up at time T D 0according to the following dynamics. There exist geometrical parameters .r.t/; �.t/;�.t// 2R�C �R

�C �R such that��

u.t; r/�1

�2p�1 .t/

ŒQe�iˇ1y ��r � r.t/

�.t/

�ei�.t/! 0 in L2.RN /: (1.13)

The speed and the radius of concentration and the phase drift are given by the asymp-totic laws:

r.t/� jt j˛1C˛ ; �.t/� jt j

11C˛ ; �.t/� jt j�

1�˛1C˛ as t " 0: (1.14)

Moreover, the blow-up speed admits the equivalent:

��ru.t/��L2�

1

.T � t /1

1C˛

as t " 0: (1.15)

Comments on the result(1) Sharp upper bound on the blow-up speed. From direct inspection by using

(1.11), the blow-up rate (1.15) of ring solutions saturates the upper bound (1.12),

�That is, with bounded kinetic energy supŒ0;T / kru.t/kL2 <C1.��If the rough profile Qe�iˇ1y in (1.13) is replaced by the precise profile of (3.32), then the convergence in(1.13) is in H1.RN / instead of L2.RN /.


which is therefore optimal in the radial setting. This shows that there is some sharp-ness in the nonlinear interpolation estimates underlying the proof of (1.12) and theassociated localized virial identity which were already at the heart of the sharp lowerbound (1.9) in [27]. We may also derive from the proof the behavior of the criticalnorm

��u.t/�� PH sc � 1

�sc .t/�

1

.T � t /sc1C˛

;

which shows as conjectured in [27] that the logarithmic lower bound (1.9) is notalways sharp, even though it is attained for the self-similar blow-up solutions builtin [29].

(2) On the restriction sc < 1. We have restricted attention in this paper to thecase sc < 1. This assumption is used to control the plain nonlinear term and ensuresthrough the energy-subcritical Cauchy theory that controllingH 1-norms is enough tocontrol the flow. We, however, conjecture that the sharp threshold for the existence ofcollapsing ring solutions is p < 5 in any dimensionN � 2. This would require exactlyas in [37] the control of higher-order Sobolev norms in the bootstrap regime corre-sponding to collapsing ring solutions exhibited in this paper. This is an independentproblem which needs to be addressed in detail.

(3) Nondispersive solutions. The construction of the ring solution relies on thestrategy to build minimal blow-up elements developed in [38]. In particular, let usstress the fact that (1.13) coupled with the laws (1.14) implies that the solution isnondispersive because

ku0kL2.RN / D kQkL2.R/

and the solution concentrates all its L2-mass at blowup:ˇ̌u.t/

ˇ̌2* kQk2

L2.R/ıxD0 as t " 0:

In fact, a three-parameter family of such minimal elements—indexed on scaling andphase invariance, and an additional internal Galilean drift parameter—is constructed.This is a major difference with [35] and [37], where the stationary ring solutionsrequire a nontrivial dispersion, and hence the full log-log machinery developed in[23] and [25]. Such minimal elements can be constructed by reintegrating the flowbackward from the singularity by using a mixed energy/virial Lyapunov functional.The key is that as observed in [38], only energy bounds on the associated linearizedoperator close to Q are required to close this analysis (see also [14], [20] for furtherillustrations). We also remark that because the problem is no longer critical, we canconstruct an approximate solution to all orders by using the slow modulated approachin [15], [23], and [38], and therefore the construction of the minimal element requires


less structure� than in [38] and the proof is particularly robust. Let us stress the factthat obtaining dispersion by using dispersive bounds for the linearized operator wouldbe particularly delicate for this problem because the leading order blow-up profile isgiven by the mass subcritical ground state for which the linearized spectrum displaysa pair of complex eigenvalues leading to oscillatory modes (see [6]). We mention thatthe existence of a minimal ring solution in the particular case N D p D 3 has beenrecently announced in [33].

(4) Arbitrary concentration of the mass. We may let the scaling symmetry (1.3)act on the solution constructed by Theorem 1.2 and obtain solutions with an arbitrarysmall or large amount of mass:ˇ̌

u.t/ˇ̌2*mıxD0 as t " 0;m > 0:

This is a spectacular difference with the mass critical problem sc D 0 where theamount of mass focused by the nonlinearity is conjectured to be quantized (see [26]).

Let us stress that Theorem 1.2 gives the first explicit description of blow-updynamics for a large set of values .N;p/, and the robust scheme behind the proofis likely to adapt to a large class of problems. One important open problem after thiswork is to understand stability properties of the collapsing ring blow-up solutions.The numeral experiments in [8] clearly indicate the stability of the ring mechanismby radial perturbation of the data, but the proof would involve dealing with dispersionnear the subcritical ground state, which is a delicate analytical problem. We moreoverexpect that the ring singularity scenario persists on suitably prepared finite codimen-sional sets of nonradial initial data.

NotationsWe introduce the differential operator

ƒf D2

p � 1f C y � rf:

Let L D .LC;L�/ be the matrix linearized operator close to the one-dimensionalground state:

LC D�@2y C 1� pQ

p�1; L� D�@2y C 1�Q

p�1: (1.16)

We recall that L has a generalized nullspace characterized by the following algebraicidentities generated by the symmetry group:

L�.Q/D 0; LC.ƒQ/D�2Q;

LC.Q0/D 0; L�.yQ/D�2Q

0:(1.17)

�Even though a similar structure could be exhibited which would probably be relevant for stability issues, andin particular, a finite order expansion is enough to close the analysis.


We note the one-dimensional scalar product

.f;g/D

Zf .y/g.y/dy:

1.4. Strategy of the proofLet us give a brief insight into the proof of Theorem 1.2. The scheme follows the roadmap designed in [38].

Step 1: A rough approximate solution. Let us renormalize the flow by using thetime-dependent rescaling

u.t; r/D1

�.t/2p�1

v�s;r � r.t/

�.t/

�ei�.t/;

ds

dtD

1

�2;

which maps the finite-time blow-up problem (1.1) onto the global-in-time renormal-ized equation� (3.4):

i@svC vyy CN � 1

1C ˛b2ˇy

˛b

2ˇvy � .1C ˇ

2/vC ibƒvC 2iˇvy C vjvjp�1

D i��s�C b

�ƒvC i

�rs�C 2ˇ

�vy C . Q�s � ˇ

2/v; (1.18)

where we have introduced a Galilean drift parameter ˇ and where we have defined

b D2ˇ

˛

�

rand Q�s D �s � 1:

The beautiful observation of Fibich, Gavish, and Wang [8] is that an approximatesolution to (1.18) can be constructed of the form

w.s;y/DQ.y/e�ib.s/y2

4 e�iˇ1y ;

where Q is the mass-subcritical one-dimensional ground state, and this relies on thespecific algebra generated by the choice (1.10) of ˇ and the specific choices of mod-ulation equations (3.5). Note that this choice corresponds to the cancellation

E.Qe�iˇ1y/D 0;

which is indeed required for a blow-up profile candidate. The explicit integration ofthe modulation equations

rs

�D�2ˇ; �

�s

�D b D

2ˇ

˛

�

r;

ds

dtD

1

�2(1.19)

with the choice ˇD ˇ1 leads from direct check to the regime (1.14) for r , �, � . Since�.t/ touches zero in finite time, this corresponds to a finite-time blow-up regime.

�Defined on y >� 2ˇ˛b

.


Furthermore, there holds the relation

b � �1�˛: (1.20)

Step 2: Construction of a high-order approximate solution. We now proceed tothe construction of a high order approximate solution to (3.4). Following the slowmodulated ansatz approach developed in [15], [23], and [38], we freeze the modula-tion equations

rs

�C 2ˇD 0; Q�s D ˇ

2; b D2ˇ

˛

�

r

and look for an expansion of the form

Qb; Q̌.y/D

hQC

X1�jCl�k�1

bj Q̌l.s/�Tj;l .y/C iSj;l.y/

�ie�i

b.s/y2

4 e�iˇy ; (1.21)

where

ˇD ˇ1C Q̌

and the laws for the remaining parameters are adjusted dynamically:

�s

�C b DP1.b; Q̌/; Q̌

s DP2.b; Q̌/:

Expanding in powers of b; Q̌, the construction reduces to an inductive linear system´LCTj;l D Fj;l.Tp;q; : : : ; Sp;q/1�pCq�jCl�1;

L�Sj;l DGj;l.Tp;q; : : : ; Sp;q/1�pCq�jCl�1;j � 1; (1.22)

where .LC;L�/ is the matrix linearized operator (1.16) close toQ. The kernel of thisoperator is well known (see [39]), and the solvability of the nonlinear system (1.22)in the class of Schwarz functions is subject to the orthogonality conditions´

.Fj;l.Tp;q; : : : ; Sp;q/1�pCq�jCl�1; @yQ/D 0;

.Gj;l.Tp;q; : : : ; Sp;q/1�pCq�jCl�1;Q/D 0;(1.23)

which correspond, respectively, to the translation and phase orbital instabilities, andis ensured inductively through the construction of the polynomials .Pi .b; Q̌//iD1;2.The fundamental observation is that the problem near the subcritical ground state isno longer degenerate, that is,

.ƒQ;Q/¤ 0;

and this is a major difference with [38] and [14]. The outcome is the construction ofan approximate solution to arbitrary high order.


Step 3: The mixed energy/Morawetz functional. We now aim at building an exactsolution and use for this the Schauder-type compactness argument designed in [21]and [18] (see also [15], [38]). We let a sequence tn " 0, and consider un.t/ the solutionto (1.1) with initial data given by the well-prepared bubble

un.tn; x/D1

�.tn/2p�1

Qb.tn/; Q̌.tn/

�r � r.tn/�.tn/

�ei�.tn/;

where the parameters are chosen in their asymptotic law (1.14):

r.tn/� jtnj˛1C˛ ; �.tn/� jtnj

11C˛ ; �.tn/� jtnj

� 1�˛1C˛ :

We then proceed to a modulated decomposition of the flow

u.t; r/D1

�.t/2p�1

.Qb.t/; Q̌.t/

C "/�t;r � r.t/

�.t/

�ei�.t/;

where " satisfies suitable orthogonality conditions through the modulation on .r.t/;�.t/; �.t/; Q̌.t//, and the b parameter is frozen:

b.t/D2ˇ.t/

˛

�.t/

r.t/:

We claim that there exists a backward time t independent of n such that

8t 2 Œt ; tn�;��".t/��

H1�� ck .t/; (1.24)

where we introduce the renormalized Sobolev norm,

k"k2H1�D

Z �j@y"j

2C j"j2��; �.y/D

�1C

˛b

2ˇy�N�1

11C˛b

2ˇy>0

;

and where ck !C1 as k!C1 relates to the order of expansion of the approx-imate solution Q

b; Q̌to (1.18). The estimate (1.24) easily allows us to conclude the

proof of existence by passing to the limit tn " 0, and the control of the parameters.�.t/; r.t// leading to concentration follows from the standard reintegration of thecorresponding modulation equation.

Following [36], [38], and [30], the proof of (1.24) relies on the derivation of amixed energy/Morawetz–Lyapunov functional. Let the Galilean shift be

Q"D "eiˇy ;

the corresponding monotonicity formula roughly takes the form

d

dtID J CO

�bk�4

�; (1.25)


where I;J are given by

I. Qu/D1

2

Zjr Quj2C

1C ˇ2

2

Zj Quj2

�2�

Z �F. QQC Qu/�F. QQ/�F 0. QQ/ � Qu

�

Cˇ

�=�Z

�� r

r.t/� 1

�@r Qu Qu

�;

with F.u/D jujpC1, � a suitable cut off function, and

J DO�bk"k2

H1�

�4

�:

The power of b in the right-hand side of (1.25) is related to the error in the construc-tion of Q

b; Q̌, and the Morawetz term in I is manufactured to reproduce the nontrivial

Galilean drift ˇ1 so that I is on the soliton core a small deformation of the linearizedenergy. Our choice of orthogonality conditions then ensures the coercivity of I:

I �k"k2

H1�

�2: (1.26)

Now, unlike in [38], we do not need to take into account further structure in thequadratic term J. Indeed, for a large enough� parameter 1, we obtain, from��t � b > 0,

d

dt

� I

��

�� b

�4C�

�. �C/k"k2

H1�

�CO

� bk

�4C�

��O

� bk

�4C�

�:

For k large enough, the last term is integrable in time in the ring regime, and inte-grating the ODE backward from blow-up time where ".tn/ 0 yields (1.24). Notethat the strength of this energy method is in particular to completely avoid the use ofweighted spaces to control the flow as in [3] and [1], and the analysis is robust enoughto handle rough nonlinearities p < 2.

This paper is organized as follows. In Section 2, we prove Theorem 1.1. In Sec-tion 3, we construct the approximate solution Q

b; Q̌by using the slowly modulated

ansatz. In Section 4, we set up the bootstrap argument and derive the modulationequations. In Section 5, we derive the mixed energy/Morawetz monotonicity formula.In Section 6, we close the bootstrap and conclude the proof of Theorem 1.2.

2. Universal upper bound on the blow-up rateThis section is devoted to the proof of Theorem 1.1. The proof is spectacularly simpleand relies on a sharp version of the localized virial identity used in [27].

�Related to the universal coercivity constant in (1.26).


Proof of Theorem 1.1Step 1: Localized virial identity. Let N � 2, 0 < sc < 1 and u 2 C.Œ0; T /;H 1/ be aradially symmetric finite-time blow-up solution 0 < T <C1. Pick a time t0 < T anda radius 0 < RDR.t0/� 1 to be chosen. Let be a radial function in D.RN /, andrecall the localized virial identity� for radial solutions:

1

2

d

d�

Zjuj2 D =

�Zr � ruu

�; (2.1)

1

2

d

d�=�Zr � ruu

�D

Z00jruj2 �

1

4

Z�2juj2 �

�12�

1

pC 1

�Z�jujpC1:

Applying with D R D R2 . xR / where is a radial function such that .x/ Djxj2

2for jxj � 2 and .x/D 0 for jxj � 3, we get

1

2

d

d�=�Zr R � ruu

�

D

Z 00� xR

�jruj2 �

1

4R2

Z�2

� xR

�juj2 �

�12�

1

pC 1

�Z�

� xR

�jujpC1

�

Zjruj2 �N

�12�

1

pC 1

�ZjujpC1

CCh 1R2

Z2R�jxj�3R

jvj2C

Zjxj�R

jujpC1i:

Now, from the conservation of the energy,ZjujpC1 D

pC 1

2

Zjruj2 � .pC 1/E.u0/;

from whichZjruj2 �N

�12�

1

pC 1

�ZjujpC1 D

N.p � 1/

2E.u0/�

2sc

N � 2sc

Zjruj2;

and thus

2sc

N � 2sc

Zjruj2C

1

2

d


�

�hjE0j C

Zjxj�R

jujpC1C1

R2

Z2R�jxj�3R

juj2i

� C.u0/h1C

1

R2C

Zjxj�R

jujpC1i

(2.2)

from the energy and L2-norm conservations.

�See [27] for further details.


Step 2: Radial Gagliardo–Nirenberg interpolation estimate. To control the outernonlinear term in (2.2), we recall the radial interpolation bound

kukL1.r�R/ �kruk

12

L2kuk

12

L2

RN�12

;

which together with the L2-conservation law ensuresZjxj�R

jujpC1 � kukp�1

L1.r�R/

Zjuj2

�C.u0/

R.N�1/.p�1/

2

krukp�12

L2

� ı2sc

N � 2sc

Zjruj2C

C

ıR2.N�1/.p�1/

.5�p/

D ı2sc

N � 2sc

Zjruj2C

C

ıR2˛

;

where we used Hölder for p < 5 and the definition of ˛ (1.11). Injecting this into(2.2) yields for ı > 0 small enough, using R� 1 and 0 < ˛ < 1,

sc

N � 2sc

Zjruj2C

d


��C.u0; p/

R2˛

: (2.3)

Step 3: Time integration. We now integrate (2.3) twice in time on Œt0; t2� byusing (2.1). This yields up to constants by using Fubini in time:Z

Rˇ̌u.t2/

ˇ̌2C

Z t2

t0

.t2 � t /��ru.t/��2

L2dt

� .t2 � t0/2

R2˛

C .t2 � t0/ˇ̌̌=�Zr R � ruu

�.t0/

ˇ̌̌C

Z Rˇ̌u.t0/

ˇ̌2

� C.u0/h .t2 � t0/2

R2˛

CR.t2 � t0/��ru.t0/��L2 CR2ku0k2L2i:

We now let t! T . We conclude that the integral in the left-hand side converges� andZ T

t0

.T � t /ˇ̌ru.t/

ˇ̌2L2dt

� C.u0/h .T � t0/2

R2˛

CR.T � t0/��ru.t0/��L2 CR2i: (2.4)

�This is consistent with (1.8) and can be proved in † without the radial assumption.


We now optimize in R by choosing

.T � t0/2

R2˛

DR2 i.e. R.t0/D .T � t0/˛1C˛ :

(2.4) now becomesZ T

t0

.T � t /��ru.t/��2

L2dt

� C.u0/�.T � t0/

2˛1C˛ C .T � t0/

˛1C˛ .T � t0/

��ru.t0/��L2�� C.u0/.T � t0/

2˛1C˛ C .T � t0/

2��ru.t0/��2L2 : (2.5)

To integrate this differential inequality, let

g.t0/D

Z T

t0

.T � t /��ru.t/��2

L2dt I (2.6)

then (2.5) means

g.t/� C.T � t /2˛1C˛ � .T � t /g0.t/;

that is, � g

T � t

�0D

1

.T � t /2

�.T � t /g0C g

��

1

.T � t /2�2˛1C˛

:

Integrating this in time yields

g.t/

T � t� C.u0/C

1

.T � t /1�2˛1C˛

i.e., g.t/� C.u0/.T � t /2˛1C˛

for t close enough to T , which together with (2.6) yields (1.12).This concludes the proof of Theorem 1.1.

3. The approximate solutionThe rest of the paper is dedicated to the proof of Theorem 1.2 on the existence of ringsolutions. We start in this section with the construction of an approximate solution atany order.

3.1. The slow modulated ansatzRecall the definition of the positive numbers ˛ and ˇ1 as

˛D5� p

.p � 1/.N � 1/; ˇ1 D

s5� p

pC 3: (3.1)


Recall also that the restrictions on p yield

0 < ˛ < 1 and 0 < ˇ1 < 1: (3.2)

Finally, recall thatQ denotes the 1-dimensional groundstate, that is, the only positive,nonzero solution in H 1 of

Q00 �QCQp D 0; explicitly Q.x/D� pC 1

2 cosh2.p�12x/

� 1p�1

: (3.3)

Let us consider the general modulated ansatz

u.t; r/D1

�.t/2p�1

v�s;r � r.t/

�.t/

�ei�.t/;

ds

dtD

1

�2;

which maps the finite-time blow-up problem (1.1) onto the global-in-time renormal-ized equation� (3.4):

i@svC vyy CN � 1

1C ˛b2ˇy

˛b

2ˇvy � .1C ˇ

2/vC ibƒvC 2iˇvy C vjvjp�1

D i��s�C b

�ƒvC i

�rs�C 2ˇ

�vy C . Q�s � ˇ

2/v; (3.4)

where we have defined

b D2ˇ

˛

�

rand Q�s D �s � 1: (3.5)

We shift a Galilean phase, and let w be defined by

w.s;y/D v.s; y/eiˇy ; (3.6)

which satisfies

i@swCwyy �wCwjwjp�1C

˛b

2ˇ

N � 1

1C ˛by2ˇ

.wy � iˇw/C b.iƒwC ˇyw/

D� Q̌sywC��s�C b

�.iƒwC ˇyw/C

�rs�C 2ˇ

�.iwy C ˇw/

C . Q�s � ˇ2/w: (3.7)

3.2. Construction of the approximate solution Qb; Q̌

We now proceed to the slow modulated ansatz construction as in [15] and [38]. Let

ˇD ˇ1C Q̌:

�Defined on y >� 2ˇ˛b

.


We look for an approximate solution to (3.4) of the form

v.s; y/DQb.s/; Q̌.s/

.y/;�s

�D�bCP1.b; Q̌/;

rs

�D�2ˇ;

Q�s D ˇ2; ˇs DP2.b; Q̌/;

where P1 and P2 are polynomial in .b; Q̌/, which will be chosen later to ensuresuitable solvability conditions. Note from the definition (3.5) of b the relation

bs C .1� ˛/b2 �

b

ˇP2 � bP1

Db

ˇ. Q̌s �P2/C b

��s�C b �P1

��˛

2ˇb2�rs�C 2ˇ

�: (3.8)

We then define the error term

�‰b; Q̌D i

��.1� ˛/b2C

b

ˇP2C bP1

�@bQb; Q̌

C iP2@ Q̌Qb; Q̌

� .1C ˇ2/Qb; Q̌C i.b �P1/

� 2

p � 1C y@y

�Qb; Q̌C 2iˇ@yQb; Q̌

C @2yQb; Q̌C˛b

2ˇ

N � 1

1C ˛b2ˇy@yQb; Q̌

C jQb; Q̌jp�1Q

b; Q̌: (3.9)

The algebra simplifies after a mixed Galilean/pseudoconformal drift:

Qb; Q̌.y/D P

b; Q̌.y/e�iˇy�ib

y2

4 ; (3.10)

which leads to the slowly modulated equation

i��.1� ˛/b2C

b

ˇP2C bP1

�@bPb; Q̌ C iP2@ Q̌Pb; Q̌

�Pb; Q̌C @2yPb; Q̌ C

˛b

2ˇ

N � 1

1C ˛b2ˇy@yPb; Q̌ C jPb; Q̌ j

p�1Pb; Q̌

� iP1

� 2

p � 1C y@y

�Pb; Q̌�P1

�ˇy C

by2

2

�Pb; Q̌CP2yPb; Q̌

C�bˇy C

�˛b2C

b

ˇP2C bP1

�y24� ih N � 11C b˛y

2ˇ

˛b2

4ˇ.1� ˛/y

i�Pb; Q̌

D�‰b; Q̌eiˇyCib

jyj2

4 : (3.11)

We now claim that we can construct a well-localized high-order approximate solutionto (3.11).


PROPOSITION 3.1 (Approximate solution)Let an integer k � 5, then there exist polynomials P1 and P2 of the form

P1.b; Q̌/DX

3�jCl�k�1

c1;j;lbj Q̌l ;

P2.b; Q̌/D�2b Q̌ CX

3�jCl�k�1

c2;j;lbj Q̌l ;

(3.12)

and smooth well-localized profiles .Tj;l ; Sj;l/1�jCl�k�1, such that

Pb; Q̌DQC

X1�jCl�k�1

bj Q̌l.Tj;l C iSj;l/ (3.13)

is a solution to (3.11) with ‰b; Q̌

smooth and well localized in y satisfying

‰b; Q̌DO

�bkjyjcke�jyj

�: (3.14)

Moreover, there holds the decay estimate

jPb; Q̌j�

�1C jyj2k

�e�jyj: (3.15)

Proof of Proposition 3.1The proof proceeds by injecting the expansion (3.13) in (3.11), identifying the termswith the same homogeneity in .b; Q̌/, and inverting the corresponding operator. Letus recall that if LD .LC;L�/ is the matrix linearized operator close to Q given by(1.16), then its kernel is explicit:

Ker¹LCº D span¹Q0º; Ker¹L�º D span¹QºI (3.16)

see [39] and [6].Step 1: General strategy. Let j C l � 1. Assume that Tp;q , c1;p;q and c2;p;q for

pC q � j C l � 1 have been constructed. Then, identifying the terms homogeneousof order .j; l/ in (3.11) yields a linear system of the following type:´

LC.Tj;l/D h1;j;l � c1;j;lˇ1yQC c2;j;lyQ;

L�.Sj;l/D h2;j;l � c1;j;lƒQ;(3.17)

where h1;j;l and h2;j;l may be computed explicitly and only depend on Tp;q , c1;p;q ,and c2;p;q for p C q � j C l � 1. The invertibility of (3.17) requires, according to(3.16), that we manufacture the orthogonality conditions .h.1/j ;Q0/D .h

.2/j ;Q/D 0;

see [36] for related issues. We also need to track the decay in space of the associatedsolution in a sharp way. We claim the following.


LEMMA 3.2For all 1� j C l � k � 1, let

c1;j;l D1

.Q;ƒQ/.h2;j;l ;Q/ and

c2;j;l D2

kQk2L2

.h1;j;l ;Q0/C

ˇ1

.Q;ƒQ/.h2;j;l ;Q/:

(3.18)

Then, there exist .Tj;l ; Sj;l/ solution of (3.17) for all 1� j C l � k�1. Furthermore,Tj;l and Sj;l are smooth and decay as

Tj;l DO�jyj2.jCl/e�jyj

�and

Sj;l DO�jyj2.jCl/e�jyj

�as y!˙1:

(3.19)

Remark 3.3Note that the quantity .Q;ƒQ/ appearing in (3.18) is given by

.Q;ƒQ/D5� p

2.p � 1/

and is well defined and not zero since 1 < p < 5. This is a major difference withrespect to the analysis in [38].

Proof of Lemma 3.2To be able to solve for .Tj;l ; Sj;l/, we need, in view of (3.16) and (3.17),

.h1;j;l � c1;j;lˇ1yQC c2;j;lyQ;Q0/D 0 and .h2;j;l � c1;j;lƒQ;Q/D 0;

which is equivalent to (3.18). Thus, choosing c1;j;l and c2;j;l as in (3.18), we maysolve for .Tj;l ; Sj;l/ solution of (3.17).

Next, we investigate the smoothness and decay properties of .Tj;l ; Sj;l/. Identi-fying the terms homogeneous of order j C l in (3.11), we have for h1;j;l and h2;j;ldefined in (3.17)8̂̂ˆ̂<ˆ̂̂̂:

h1;j;l DPpCq�jCl�1.a1;p;qy

jCl�p�qTp;q C a2;p;qyjCl�p�qSp;q

C a3;p;qyjCl�p�qT 0p;q/CNL.1/j ;

h2;j;l DPpCq�jCl�1.a4;p;qy

jCl�p�qTp;q C a5;p;qyjCl�p�qSp;q

C a6;p;qyjCl�p�qT 0p;q/CNL.2/j ;

(3.20)

where we have defined by convenience T0;0 D Q, where am;p;q are real numberswhich may be explicitly computed, and where NL.1/j and NL.2/j are the contributions


coming from the Taylor expansion of the nonlinearity nearQ. They take the followingform:

NL.1/j DX

p�0;q�0

Xjm�1;lm�1=j1C��CjqCl1C��ClqDjCl

a.1/

j1;:::;jq ;l1;:::;lq

� Tj1;l1 � � �Tjp ;lpSjpC1;lCpC1 � � �Sjq ;lqQp�j�l ; (3.21)

and

NL.2/j DX

p�0;q�0

Xjm�1;lm�1=j1C��CjqCl1C��ClqDjCl

a.2/

j1;:::;jq ;l1;:::;lq

� Tj1;l1 � � �Tjp ;lpSjpC1;lCpC1 � � �Sjq ;lqQp�j�l ; (3.22)

where the real numbers a.1/j1;:::;jq ;l1;:::;lq

and a.2/j1;:::;jq ;l1;:::;lq

may be computed explic-itly.

We argue by induction. Assume that Tp;q , pC q � j C l � 1 satisfy the conclu-sions of the lemma in terms of smoothness and decay. Then, we easily check fromthe formulas (3.20), (3.21), and (3.22) that h1;j;l and h2;j;l are smooth. Then, fromstandard elliptic regularity, we deduce that Tj;l and Sj;l are smooth.

Finally, we consider the decay properties of Tj;l and Sj;l . Since we assume byinduction that Tp;q , pCq � j C l �1, satisfy the decay assumption (3.19), we easilyobtain from (3.21) and (3.22)

NL.1/j DO�jyj2.jCl/e�pjyj

�and

NL.2/j DO�jyj2.jCl/e�pjyj

�as y!˙1:

Together with (3.17), the fact that Tp;q , pCq � jCl�1 satisfy the decay assumption(3.19), and the fact that p > 1, we deduce

h1;j;l DO�jyj2j�1e�jyj

�and

h2;j;l DO�jyj2j�1e�jyj

�as y!˙1:

(3.23)

Now, let us consider the solution .f1; f2/ to the system

LC.f1/D h1 and L�.f2/D h2;

with .h1;Q0/D 0 and .h2;Q/D 0. Then, for y � 1 for instance, we define

g1.y/DQ0.y/

Z y

1

d�

Q0.�/2and g2.y/DQ.y/

Z y

1

d�

Q.�/2;

so that .Q0; g1/ forms a basis of solutions to the second-order ordinary differentialequation LC.f / D 0, while .Q;g2/ forms a basis of solutions to the second-order


ordinary differential equation L�.f /D 0. Note that the decay properties of Q andQ0 immediately yield

g1.y/DO.ey/ and g2.y/DO.e

y/ as y!C1:

Furthermore, using the variation of constants method, we find that a solution is givenby�

f1.y/D�g1.y/�Z C1y

h1.�/Q0.�/d�

�CQ0.y/

�Z y

1

h1.�/g1.�/d��;

f2.y/D�g2.y/�Z C1y

h2.�/Q.�/d��CQ.y/

�Z y

1

h2.�/g2.�/d��:

Thus, for any integer n, if

h1 DO�jyjne�jyj

�and h2.y/DO

�jyjne�jyj

�as y!˙1;

then we obtain

f1 DO�jyjnC1e�jyj

�and f2.y/DO

�jyjnC1e�jyj

�as y!˙1:

Applying this observation to the system (3.17) with the choice nD 2.jCl/�1 yields,in view of (3.23), the decay (3.19). This concludes the proof of the lemma.

In view of Lemma 3.2, the proof of Proposition 3.1 will follow from the verifica-tion that

cn;1;0 D cn;0;1 D cn;2;0 D cn;0;2 D 0 for nD 1; 2;

c1;1;1 D 0 and c2;1;1 D�2:

Step 2: Computation of c1;1;0 and c2;1;0. We identify the terms homogeneous oforder .1; 0/ in (3.11) and get´

LC.T1;0/D.N�1/˛2ˇ1

Q0C ˇ1yQ� c1;1;0ˇ1yQC c2;1;0yQ;

L�.S1;0/D�c1;1;0ƒQ:(3.24)

Now, note that� .N � 1/˛2ˇ1

Q0C ˇ1yQ;Q0�D.N � 1/˛

2ˇ1

Z.Q0/2 �

ˇ1

2

ZQ2

Dˇ1

2

�pC 3p � 1

Z.Q0/2 �

ZQ2�; (3.25)

�Note that solutions are given up to an element of the kernel, but adjusting this element is irrelevant.


where we used in the last inequality the definition of ˛ and ˇ1 given by (1.10) and(1.11). Now, taking the scalar product of (3.3) with QC .pC 1/yQ0 and integratingby parts, yields Z

Q2 DpC 3

p � 1

Z.Q0/2; (3.26)

which together with (3.25) implies� .N � 1/˛2ˇ1

Q0C ˇ1yQ;Q0�D 0:

Together with (3.24), we obtain

.h1;1;0;Q0/D .h2;1;0;Q/D 0;

which together with (3.18) yields

c1;1;0 D c2;1;0 D 0

as desired.Step 3: Computation of c1;0;1 and c2;0;1. We identify the terms homogeneous of

order .1; 0/ in (3.11) and get´LC.T0;1/D�c1;0;1ˇ1yQC c2;0;1yQ;

L�.S0;1/D�c1;0;1ƒQ;(3.27)


c1;0;1 D c2;0;1 D 0


order .2; 0/ in (3.11) and get8̂̂ˆ̂̂̂̂<ˆ̂̂̂̂ˆ̂:

LC.T2;0/D .1� ˛/S1;0C.N�1/˛2ˇ1

T 01;0 � .N � 1/˛2

4ˇ21yQ0

C p.p�1/2

Qp�2T 21;0Cp�12Qp�2S21;0C ˇ1yT1;0

C ˛4y2Q� c1;2;0ˇ1yQC c2;2;0yQ;

L�.S2;0/D�.1� ˛/T1C.N�1/˛2ˇ1

S 01;0C .p � 1/Qp�2T1;0S1;0

C ˇ1yS1;0 � .N � 1/˛

4ˇ1yQ.1� ˛/� c1;2;0ƒQ:

(3.28)

Note from (3.24) that T1;0 is an odd function, while S1;0 is an even function. In viewof (3.28), this implies that

h1;2;0 is even and h2;2;0 is odd:


In particular, since Q is even and Q0 is odd, we obtain

.h1;2;0;Q0/D 0 and .h2;2;0;Q/D 0;


c1;2;0 D c2;2;0 D 0


order .1; 1/ in (3.11) and get´LC.T1;1/D�

.N�1/˛

2ˇ21Q0C yQ� c1;1;1ˇ1yQC c2;1;1yQ;

L�.S1;1/D�c1;1;1ƒQ:(3.29)

In view of (3.29), we have

.h1;1;1;Q0/D�

.N � 1/˛

2ˇ21

Z.Q0/2 �

1

2

ZQ2:

Using the computation Z.Q0/2 D

p � 1

pC 3

ZQ2

and the definition of ˛ and ˇ1 given by (1.10) and (1.11), we deduce

.h1;1;1;Q0/D�

ZQ2;

which together with (3.18) and the fact that h2;1;1 D 0 yields

c1;1;1 D 0 and c2;1;1 D�2


order .0; 2/ in (3.11) and get´LC.T0;2/D�c1;0;2ˇ1yQC c2;0;2yQ;

L�.S0;2/D�c1;0;2ƒQ;(3.30)


c1;0;2 D c2;0;2 D 0

as desired.


Step 7: Conclusion. We therefore have constructed an approximate solution Pb; Q̌

of (3.11) of the form (3.13). The decay estimate (3.15) on Pb; Q̌

follows from (3.19).The error ‰

b; Q̌consists of a polynomial in .Tj;l ; Sj;l/jCl�k�1 with lower-order k,

the error between the Taylor expansion of the potential terms ˛b2ˇ

N�1

1C˛b2ˇy

and N�1

1C b˛y2ˇ

in

(3.11), and the error between the nonlinear term and its Taylor expansion. The first andsecond type of terms are easily treated using the uniform exponential decay of P

b; Q̌.

We need to be slightly more careful for the nonlinear term. Here we recall that givenz 2C, let Pk�1.z/ be the order k � 1 Taylor polynomial of z 7! .1C z/j1C zjp�1 atz D 0; then from� p < 5� k,

8z 2C;ˇ̌.1C z/j1C zjp�1 �Pk�1.z/

ˇ̌� Ckjzjk :

Then let

"b; Q̌D P

b; Q̌�Q;

and we obtain the bound by homogeneity:ˇ̌̌.QC "

b; Q̌/jQC "

b; Q̌jp�1 �QpPk�1

�"b; Q̌

Q

�ˇ̌̌

� CkQpj"b; Q̌jk

Qk

�QpX

1�jCl�k�1

� jbjj j Q̌jl.jTj;l j C jSj;l j/Q

�k:

On the other hand, (3.19) ensures the uniform bound

jTj;l j C jSj;l j

Q� 1C jyjck ; j C l � k � 1;

and hence the boundˇ̌̌.QC "b/jQC "bj

p�1 �QpPk�1

�"bQ

�ˇ̌̌��jbj C j Q̌j

�kjyjcke�jyj;

and the control (3.14) of ‰b; Q̌

follows.This concludes the proof of Proposition 3.1.

3.3. Further properties of Qb; Q̌

To avoid artificial troubles near the origin after renormalization, we introduce asmooth cutoff function

�To handle the case when jzj� 1.


�.y/D

´0 for y ��2;

1 for y ��1;�b.y/D �.

pby/; (3.31)

and define, once and for all for the rest of this paper,

Qb; Q̌.y/D �b.y/Pb; Q̌.y/e

�iˇy�ib y2

4 : (3.32)

Let us rewrite the Qb; Q̌

equation by using (3.4) and (3.9) in the form which we willuse in the forthcoming bootstrap argument.

COROLLARY 3.4 (Qb; Q̌

equation in original variables)

Given C1 modulation parameters .�.t/; r.t/; �.t/; Q̌.t// such that

0 < b.t/D2ˇ

˛

�.t/

r.t/� 1; (3.33)

let QQ be given by

QQ.t; x/D1

�2p�1

Qb.t/; Q̌.t/

�r � r.t/�.t/

�ei�.t/: (3.34)

Then QQ is a smooth radially symmetric function which satisfies

i@t QQC� QQC QQj QQjp�1 D D

1

�.t/2pp�1

‰�t;r � r.t/

�.t/

�ei�.t/ (3.35)

with

‰ D�.�s � 1� ˇ2/Q

b; Q̌� i��s�C b �P1

�.ƒQ

b; Q̌� b@bQb; Q̌

/

� i�rs�C 2ˇ

��@yQb; Q̌

C˛

2ˇb2@bQb; Q̌

�

C i�bs C .1� ˛/b

2 �b

ˇP2 � bP1

�@bQb; Q̌

C i.ˇs �P2/�@ Q̌Qb; Q̌

Cb

ˇ@bQb; Q̌

�

CO�e�jyjbck

1y� 1pb

C�jbj C j Q̌j

�k�bjyj

cke�jyj�: (3.36)

Proof of Corollary 3.4We simply observe from (3.32) and (3.33) that QQ is identically zero near the originand hence (3.34) defines a well-localized smooth radially symmetric function. The


exponential decay in space of Pb; Q̌

ensures that the localization procedures perturb

the error term in (3.11) by an O. e�jyj

bck1y� 1p

b

/ and the estimate (3.36) now directly

follows from (3.4), (3.8), (3.9), and (3.14).

4. Setting up the analysisThe aim of this section is to set up the bootstrap argument.

4.1. Choice of initial dataLet us start with solving the system of exact modulation equations formally predictedby the Q

b; Q̌construction. It is easily seen that this system formally predicts a stable

blowup in the ring regime. We shall simply need the following claim, whose proof iselementary and postponed until Appendix A.

LEMMA 4.1 (Integration of the exact system of modulation equations)There exists te < 0 small enough and a solution .�e; be; Q̌e; re; �e/ to the dynamicalsystem 8̂̂

ˆ̂̂̂̂ˆ̂<ˆ̂̂̂̂ˆ̂̂̂:

�s�C b DP1.b; Q̌/;

rs�C 2ˇD 0;

Q̌s DP2.b; Q̌/;

�s D 1C ˇ2;

dsdtD 1

�2;

b D 2ˇ˛�r; ˇD ˇ1C Q̌;

(4.1)

which is defined on Œte; 0/. Moreover, this solution satisfies the following bounds,

be.t/D1

1C ˛

�2.1C ˛/ˇ1˛g1

� 21C˛jt j

1�˛1C˛

�1CO

�log�jt j�jt j

1�˛1C˛

��; (4.2)

�e.t/D�2.1C ˛/ˇ1

˛g1

� 11C˛jt j

11C˛

�1CO


1�˛1C˛

��; (4.3)

re.t/D g1

�2.1C ˛/ˇ1˛g1

� ˛1C˛jt j

˛1C˛

�1CO


1�˛1C˛

��; (4.4)

Q̌e.t/DO

�jt j

2.1�˛/1C˛

�; (4.5)

and

�e.t/D .1C ˇ21/� 1� ˛1C ˛

� 1�˛1C˛

�2.1� ˛/ˇ1˛g1

�� 21C˛jt j�

1�˛1C˛ CO

�log�jt j��; (4.6)

for some universal constant jg1 � 1j � 1.


From now on, we choose the integer k appearing in Proposition 3.1 such that

k >2

1� ˛C 1: (4.7)

Given te < Nt < 0 small, we let u.t/ be the solution to (1.1) with well-prepared initialdata at t D Nt given explicitly by

u.Nt ; r/D1

�e.Nt /2p�1

Qbe.Nt /; Q̌e.Nt /

�r � re.Nt /�e.Nt /

�ei�e.

Nt /: (4.8)

Our aim is to derive bounds on u backward on a time interval independent of Nt asNt ! 0. We describe in this section the bootstrap regime in which we will control thesolution, and we derive preliminary estimates on the flow which prepare the mono-tonicity formula of Section 5.

4.2. The modulation argumentWe prove in this section a standard modulation lemma which relies on the implicitfunction theorem and the mass subcritical nondegeneracy .Q;ƒQ/¤ 0.

LEMMA 4.2 (Modulation)There exists a universal constant ı > 0 such that the following holds. Let u be aradially symmetric function of the form

u.r/D1

�2p�1

0

Qb0; Q̌0

�r � r0�0

�ei�0 C Qu0.r/

with

�0; r0 > 0; ˇ0 D ˇ1C Q̌0; b0 D2ˇ0

˛

�0

r0;

the a priori bound

r0

�˛0� 1; (4.9)

and the a priori smallness

0 < jb0j C j Q̌0j C k Qu0kL2 < ı: (4.10)

Then there exists a unique decomposition

u.t; r/D1

�2p�1

1

Qb1; Q̌1

�r � r1�0

�ei�1 C Qu1.r/


with

ˇ1 D ˇ1C Q̌1; b1 D2ˇ1

˛

�1

r1;

such that

Qu1.r/D1

�2p�1

1

Q"1

�r � r1�1

�ei�1�iˇ1

r�r1�1

satisfies the orthogonality conditions�<.Q"1/; �b1yQ

�D�<.Q"1/; �b1Q

�D�=.Q"1/; �b1ƒQ

�D�=.Q"1/; �b1@yQ

�D 0:

Moreover, there holds the smallnessˇ̌̌�1�0� 1

ˇ̌̌Cjr0 � r1j

�0C j Q̌0 � Q̌1j C j�0 � �1j C k Qu1kL2 � ı: (4.11)

Proof of Lemma 4.2This is a standard consequence of the implicit function theorem. We have, by assump-tion,

u.r/D1

�2p�1

0

Qb0; Q̌0

�r � r0�0

�ei�0 C Qu0.r/;

and we wish to introduce a modified decomposition

u.r/D1

�2p�1

1

Qb1; Q̌1

�r � r1�1

�ei�1 C Qu1.r/:

Comparing the decompositions, we obtain the formula

Qu1.r/D1

�2p�1

0

Qb0; Q̌0

�r � r0�0

�ei�0 �

1

�2p�1

1

Qb1; Q̌1

�r � r1�1

�ei�1 C Qu0.r/:

We now form the functional

Fz;�;�; Q̌

.y/D �2p�1Q

b0; Q̌0.�y C z/e�i�Ci.ˇ0C

Q̌/y �Qb1; Q̌1

.y/eiˇ1y (4.12)

with

z Dr1 � r0

�0; �D

�1

�0; � D �1 � �0; Q̌ D Q̌1 � Q̌0;

so that

Q"1.y/D Fz;�;�; Q̌.y/C �2p�1

1 Qu0.�1y C r1/e�i�1Ciˇ1y :


We then define the scalar products

.j / D

Z C1�1

<.Q"1/�b1T.j / dy

D

Z C1�1

<.Fz;�;�; Q̌

/�b1T.j / dy

C<�Z C10

Qu0.r/�

2p�1

1

�1.�b1ƒQ/

�r � r1�1

�e�i�1Ciˇ

r�r1�1 dr

�for j D 1; 2;

and

.j / D

Z C1�1

=.Q"1/�b1T.j / dy

D

Z C1�1

=.Fz;�;�; Q̌

/�b1T.j / dy

C=�Z C10

Qu0.r/�

2p�1

1

�1.�b1ƒQ/

�r � r1�1

�e�i�1Ciˇ

r�r1�1 dr

�for j D 3; 4;

where

T .1/ D yQ; T .2/ DQ; T .3/ D @yQ; T .4/ DƒQ:

We now view D . .j //1�j�4 as smooth functions of . Qu0; z;�; Q̌; �/. Observe thatthe bound (4.9) ensures using the explicit formula (1.11) for ˛:

ˇ̌ . Qu0; 0; 1; 0; 0/

ˇ̌�� r0

�5�p

.N�1/.p�1/

0

��N�12k Qu0kL2 � ı: (4.13)

We now compute

b1 D2ˇ1

˛

�1

r1D 2.ˇ0C Q̌/

�0

˛r0�r0

r1D�1C

Q̌

ˇ0

�b0�

�1C

˛b0

2ˇ0z��1

:

We thus obtain using

.Qb; Q̌/j.b; Q̌/D.0;0/

DQe�iˇ1y

the infinitesimal deformations

@zFj.zD0;�D1; Q̌D0;�D0/

DQ0 � iˇ1QCO��jb0j C j Q̌0j

�e�cjyj

�;

@�Fj.zD0;�D1; Q̌D0;�D0/

DƒQ� iˇ1yQCO��jb0j C j Q̌0j

�e�cjyj

�;

@ Q̌Fj.zD0;�D1; Q̌D0;�D0/

D iyQCO��jb0j C j Q̌0j

�e�cjyj

�;

@�Fj.zD0;�D1; Q̌D0;�D0/

D �iQCO��jb0j C j Q̌0j

�e�cjyj

�:


The Jacobian matrix of at . Qu0 D 0; z D 0;�D 1; Q̌ D 0; � D 0/ is therefore givenby

D D

ˇ̌̌ˇ̌̌ˇ̌.Q0; yQ/ .ƒQ;yQ/ 0 0

.Q0;Q/ .ƒQ;Q/ 0 0

�ˇ1.Q;Q0/ �ˇ1.yQ;Q

0/ .yQ;Q0/ �.Q;Q0/

�ˇ1.Q;ƒQ/ �ˇ.yQ;ƒQ/ .yQ;ƒQ/ �.Q;ƒQ/

ˇ̌̌ˇ̌̌ˇ̌CO

�jb0j C j Q̌0j

�

D �1

16

�5� pp � 1

�2kQk8

L2CO

�jb0j C j Q̌0j

�¤ 0

from the smallness assumption (4.10). The existence of the desired decompositionnow follows from the implicit function theorem, and the bound (4.11) followsfrom (4.13).

4.3. Setting up the bootstrapLet u.t; r/ be the radially symmetric solution emanating from the data (4.8) at t D Nt .From Lemma 4.1, Lemma 4.2, and a straightforward continuity argument, we can finda small time t� < Nt such that u.t; r/ admits on Œt�; Nt � a unique decomposition

u.t; r/D1

�.t/2p�1

v�t;r � r.t/

�.t/

�ei�.t/; (4.14)

where we froze the law

b.t/D2ˇ

˛

�

r;

ds

dtD

1

�2.t/; (4.15)

and where there holds the decomposition

w.s;y/D v.s; y/eiˇy DQb.t/; Q̌.t/

eiˇy C Q".t; y/; Q"D Q"1C i Q"2 (4.16)

with the orthogonality conditions

.Q"1; �byQ/D .Q"1; �bQ/D .Q"2; �bƒQ/D .Q"2; �b@yQ/D 0: (4.17)

Let us define the renormalized weight on the Lebesgue measure,

�D�1C

�.t/

r.t/y�N�1

D�1C

˛b

2ˇy�N�1

; (4.18)

and the weighted Sobolev norms,

k"k2L2�D

Zj"j2�; k"k2

H1�D

Zj@y"j

2�C

Zj"j2�I

then from Lemma 4.2, the decomposition (4.14) holds as long as


r.t/

�.t/˛� 1

and ˇ̌b.t/

ˇ̌Cˇ̌Q̌.t/

ˇ̌C��Q".t/��

L2�< ı

for some universal constant ı > 0 small enough.We also introduce the decomposition of the flow:

u.t; r/D QQ.t; x/C Qu.t; r/; Qu.t; r/D1

�.t/2p�1

"�t;r � r.t/

�.t/

�ei�.t/ (4.19)

and thus

Q".s; y/D ".s; y/eiˇy : (4.20)

From (4.8), we have the well-prepared data initialization

".Nt /D 0; .�; b; Q̌; r; �/.Nt /D .�e; be; Q̌e; re; �e/.Nt /;

and we may thus consider a backward time t < Nt such that the following bootstrapassumptions hold 8t 2 .t ; Nt �:

k"kH1� < min.b;�/ı; (4.21)

0 < b < ı; (4.22)

j Q̌j � b32 ; (4.23)

and

g1

2�r.t/

�.t/˛� 2g1: (4.24)

In particular, the modulation decomposition of Lemma 4.2 applies. Our claim is thatthe above regime is trapped.

PROPOSITION 4.3 (Improvement of the bootstrap assumptions)Let te be as defined in Lemma 4.1. For any t such that te � t < Nt , and such that thebootstrap assumptions (4.21)–(4.24) are satisfied on .t ; Nt �, there holds 8t 2 .t ; Nt �,

k"kH1� � min�jt j

11C˛ ; �

�jt j

11C˛ ; (4.25)

b D1

1C ˛

�2.1C ˛/ˇ1˛g1

� 21C˛jt j

1�˛1C˛

�1CO


1�˛1C˛

��; (4.26)

j Q̌j � jt j2.1�˛/1C˛ (4.27)


and

r.t/

�.t/˛D g1

�1CO


1�˛1C˛

��: (4.28)

Proposition 4.3 is the heart of the proof of Theorem 1.2 and relies on a refinementof the energy method designed in [38].

We finish this section by deriving preliminary estimates on the decomposition(4.19) which are mostly a consequence of the construction of Q

b; Q̌and the choice of

orthogonality conditions (4.17). These estimates prepare the monotonicity formula ofSection 5, which is the key ingredient of the proof.

4.4. Modulation equationsWe derive the modulation equations associated to the modulated parameters .�.t/;r.t/; Q̌.t/; �.t//. The parameter b is computed from (4.15), which yields

bs C .1� ˛/b2 �

b

ˇP2 � bP1

Db

ˇ. Q̌s �P2/C b

��s�C b �P1

��˛

2ˇb2�rs�C 2ˇ

�: (4.29)

The modulation equations are a consequence of the orthogonality conditions (4.17)and require the derivation of the equation for Q". Recall the equation (3.7) satisfiedby w:

i@swCwyy �wCwjwjp�1C

˛b

2ˇ

N � 1

1C ˛by2ˇ

.wy � iˇw/C b.iƒwC ˇyw/

D� Q̌sywC��s�C b

�.iƒwC ˇyw/C

�rs�C 2ˇ

�.iwy C ˇw/C . Q�s � ˇ

2/w:

We inject the decomposition (4.16), which we rewrite using (3.32):

wD �bPb; Q̌e�ib y

2

4 C Q":

We then define

Mod.t/Dˇ̌̌rs�C 2ˇ

ˇ̌̌C j Q�s � ˇ

2j Cˇ̌̌�s�C b �P1

ˇ̌̌C j Q̌s �P2j;

and we obtain, using (4.29), the formula (3.36), and the fact that Pb; Q̌D Q C

O.be�cjyj/, the following system of equations for Q"1; Q"2:

@s Q"1 �L�.Q"2/

D�˛b

2ˇ

.N � 1/

1C ˛b2ˇy

�.Q"2/y � ˇ Q"1

�� Q̌sy Q"2C

��s�C b �P1

�ƒQ


C�s

�.ƒQ"1C ˇy Q"2/C

�rs�C 2ˇ

��Qy C .Q"1/y

�C � Q"2

�=R.Q"/CO��bj Q"j C bk C bMod

�e�cjyj

�; (4.30)

and

@s Q"2CLC.Q"1/

D�˛b

2ˇ

.N � 1/

1C ˛b2ˇy

��.Q"1/y � ˇ Q"2

�C . Q̌s �P2/yQC Q̌sy Q"1

� ˇ��s�C b �P1

�yQC

�s

�.ƒQ"2 � ˇy Q"1/C

�rs�C 2ˇ

�.Q"2/y

� �.QC Q"1/C<R.Q"/CO��bj Q"j C bk C bMod

�e�cjyj

�; (4.31)

where

� D . Q�s � ˇ2/C ˇ

�rs�C 2ˇ

�; (4.32)

LC and L� are the matrix linearized operator close to Q,

LC D�@2y C 1� pQ

p�1; L� D�@2y C 1�Q

p�1; (4.33)

and the nonlinear term is given by

R.Q"/D f .Qb; Q̌eiˇy C Q"/� f .Q

b; Q̌eiˇy/� f 0.Q

b; Q̌eiˇy/ � Q"

with

f .u/D ujujp�1: (4.34)

We are now in position to derive the modulation equations.

LEMMA 4.4 (Modulation equations)There holds the bounds

Mod � bkQ"kH1� C bk; (4.35)ˇ̌̌

bs C .1� ˛/b2 �

b

ˇP2 � bP1

ˇ̌̌� b2kQ"kH1� C b

kC1: (4.36)

Proof of Lemma 4.4We multiply the equation of Q"1 (4.30) by �byQ and integrate. Using the orthogonalityconditions (4.17), the identity L�.yQ/D�2Q0, and the nondegeneracy


.@yQ;�byQ/D�1

2kQk2

L2CO.e

� cpb /; (4.37)

we obtainˇ̌̌rs�C 2ˇ

ˇ̌̌� bkQ"kL2� CMod

�bCkQ"kL2�

�C bk C

ZjyjC

ˇ̌R.Q"/

ˇ̌�be�jyj: (4.38)

Next, we multiply the equation of Q"2 (4.31) by �bƒQ and use the orthogonality con-ditions (4.17), the identity LC.ƒQ/D�2Q, and the nondegeneracy

.�bƒQ;Q/D5� p

2.p � 1/

�ZQ2�CO.e

� cpb /¤ 0 (4.39)

to compute

j�j� bkQ"kL2� CMod�bCkQ"kL2�

�C bk C

ZjyjC

ˇ̌R.Q"/

ˇ̌�be�jyj: (4.40)

Next, we multiply the equation of Q"1 (4.30) by �bQ and integrate. Using the orthog-onality condition (4.17), the identity L�.Q/D 0, and the nondegeneracy (4.39), weobtain ˇ̌̌�s

�C b �P1

ˇ̌̌

� bkQ"kL2� CMod�bCkQ"kL2�

�C bk C

ZjyjC

ˇ̌R.Q"/

ˇ̌�be�jyj: (4.41)

Finally, we multiply the equation of Q"2 (4.31) by �bQ0 and use the orthogonalitycondition (4.17), the identity LC.Q0/D 0, and the nondegeneracy (4.37), to obtain

j Q̌s �P2j� bkQ"kL2� CMod�bCkQ"kL2�

�C bk C

ZjyjC

ˇ̌R.Q"/

ˇ̌�be�jyj: (4.42)

In order to estimate the nonlinear term, we first use the one-dimensional Sobolev�

k"kL1.y�� ıb/ � k"

0k12

L2.y�� ıb/k"k

12

L2.y�� ıb/� k"kH1� : (4.43)

We then estimate by direct inspection,��

8z 2C;ˇ̌f .1C z/� f .1/� f 0.1/z

ˇ̌� jzj2C jzjp1p>2; (4.44)

and hence by homogeneity,ˇ̌R.Q"/

ˇ̌� jQ

b; Q̌jp�2j"j2C j"jp1p>2: (4.45)

�Recall that �D .1C ˛b2ˇy/N�1 and thus, y >� ı

bimplies �� 1.

��Let us recall that p > 1 but p < 2 is allowed in our range of parameters.


We therefore conclude from the decay (3.15) thatZjyjC

ˇ̌R.Q"/

ˇ̌�be�jyj �

ZjyjCk�

p�1

be�.p�1/jyjj"j2C 1p>2

Zj"jp�b

� k"k2L2�C 1p>2k"k

p�2L1 k"k

2

L2�� k"k2

L2�;

where we used the Sobolev bound (4.43) and the bootstrap bound (4.21) in the laststep. Injecting this estimate into (4.38), (4.40), (4.41), and (4.42) yields (4.35). Now(4.36) follows from (4.35) and (4.29).

5. Monotonicity formulaWe now turn to the core of our analysis, which is the derivation of a monotonicityformula for the norm of " that relies on a mixed energy/Morawetz functional in thecontinuation of [36] and [38]. As in [38], the required repulsivity properties for thelinearized operator are thanks to the minimal mass assumption energy bounds onlywhich are well known for the mass-subcritical ground state. The additional Morawetzterm is designed to produce the expected nontrivial Galilean drift on the soliton coreafter renormalization.

5.1. Algebraic identityWe recall the decomposition (4.19), which in view of (3.35), yields the equation for Qu,

i@t QuC� QuC jujp�1u� QQj QQjp�1 D� D�

1

�.t/2pp�1

‰�t;r � r.t/

�.t/

�ei�.t/ (5.1)

with ‰ given by (3.36). We let

� W Œ�1;C1/!R

be a time-independent smooth compactly supported cutoff function which satisfies

�.z/ 0 for �1� z ��1

2and for z �

1

2; (5.2)

and

�.0/D 1; supz��1

ˇ̌�.z/

ˇ̌<

p1C ˇ21ˇ1

: (5.3)

Let

F.u/D1

pC 1jujpC1; f .u/D ujujp�1 so that F 0.u/ � hD<

�f .u/h

�:

We first claim a purely algebraic identity for the linearized flow (5.1) which is a mixedenergy/Morawetz functional.


LEMMA 5.1 (Algebraic energy/Morawetz estimate)Let

I. Qu/D1

2

Zjr Quj2C

1C ˇ2

2

Zj Quj2

�2�

Z �F. QQC Qu/�F. QQ/�F 0. QQ/ � Qu

�

Cˇ

�=�Z

�� r

r.t/� 1

�@r Qu Qu

�; (5.4)

J. Qu/D �1C ˇ2

�2=�f .u/� f . QQ/; Qu

��2ˇ

�<�Z

�� r

r.t/� 1

��f . QQC Qu/� f . QQ/

�@r Qu

�

�<�@t QQ;

�f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

��I (5.5)

then there holds

d

dtI. Qu/D J. Qu/CO

� b�4k"k2

H1�Cbk

�4k"kH1�

�: (5.6)

Proof of Lemma 5.1Step 1: Algebraic derivation of the energetic part. We compute from (5.1):

d

dt

°12

Zjr Quj2C

1C ˇ2

2

Zj Quj2

�2�

Z ��F.u/�F. QQ/�F 0. QQ/ � Qu

��±

D�<�@t Qu;� Qu�

1C ˇ2

�2QuC

�f .u/� f . QQ/

��.1C ˇ2/�t

�3

Zj Quj2

Cˇˇt

�2

Zj Quj2 �<

�@t QQ;

�f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

��

D=� ;� Qu�

1C ˇ2

�2QuC

�f .u/� f . QQ/

��1C ˇ2

�2=�f .u/� f . QQ/; Qu

��.1C ˇ2/�t

�3

Zj Quj2C

ˇˇt

�2

Zj Quj2

�<�@t QQ;

�f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

��: (5.7)

We first estimate from (4.35):

��t

�3

Zj Quj2 D

b

�4

Zj Quj2 �

P1

�4

Zj Quj2 �

1

�4

��s�C b

�k Quk2

L2

D1

�4O�bk"k2

L2�

�; (5.8)


where we used the bootstrap assumptions (4.21), (4.23), and (4.24) in the last equality.Also, using again (4.35), we have

ˇˇt

�2

Zj Quj2 D

ˇP2

�4

Zj Quj2C

ˇ.ˇs �P2/

�4

Zj Quj2

D1

�4O�bk"k2

L2�

�; (5.9)

where we used the bootstrap assumptions (4.21), (4.22), (4.23), and (4.24) in the lastequality.

It remains to estimate the first term in the right-hand side of (5.7). We have

ˇ̌̌=� ;� Qu�

1C ˇ2

�2QuC

�f .u/� f . QQ/

��1C ˇ2

�2=�f .u/� f . QQ/; Qu

�ˇ̌̌

�ˇ̌̌=�Z h

� � .1C ˇ2/

�2CpC 1

2j QQjp�1 �

p � 1

2j QQjp�3 QQ2

iQu�ˇ̌̌

Cˇ̌=� ;�f . QQC Qu/� f . QQ/� f 0. QQ/ � Qu

��ˇ̌:

We extract from (3.36), (4.35), and (4.36) the bound

j‰j � �b.bk CMod/

�1C jyjck

�e�jyjC

e�jyj

bck1y� 1p

b

� �b�bk C bk"kH1�

��1C jyjck

�e�jyjC

e�jyj

bck1y� 1p

b

: (5.10)

Then, we estimate in brute force:ˇ̌̌=�Z h

� � .1C ˇ2/

�2CpC 1

2j QQjp�1 �

p � 1

2j QQjp�3 QQ2

iQu�ˇ̌̌

�.bk C bk"kH1�/k"kH1�

�4:

Also, we estimate using the homogeneity estimate (4.45):ˇ̌=� ;�f . QQC Qu/� f . QQ/� f 0. QQ/ � Qu

��ˇ̌� 1

�4

Z h�b�bk C bk"kH1�

��1C jyjck

�e�jyjC

e�jyj

bck1y� 1p

b

i��jQ

b; Q̌jp�2j"j2C j"jp1p>2

�� b

�4k"k2

H1�;

where we used the Sobolev bound (4.43) in the last step. We have therefore obtainedthe preliminary computation


d

dt

°12

Zjr Quj2C

1C ˇ2

2

Zj Quj2

�2�

Z ��F.u/�F. QQ/�F 0. QQ/ � Qu

��±

D�1C ˇ2

�2=�f .u/� f . QQ/; Qu

��<

�@t QQ;

�f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

��C

1

�4O�bkk"kH1� C bk"k

2

H1�

�: (5.11)

Step 2: Algebraic derivation of the localized virial part. We now estimate thecontribution of the localized Morawetz term. We first compute using (4.15):

d

dt

h r

r.t/

iD�

rt .t/r

r2.t/

D�rs

�

r

�r2.t/

D2ˇr

�.t/r.t/2��rs�C 2ˇ

� r

�.t/r2.t/

D˛b.t/

�2.t/

r

r.t/�

˛b

2ˇ�.t/2r

r.t/

�rs�C 2ˇ

�:

This yields

d

dt

°ˇ�=�Z

�� r

r.t/� 1

�@r Qu Qu

�±

D˛ˇb

�3=�Z r

r.t/�0� r

r.t/� 1

�@r Qu Qu

�

�˛b

2�3

�rs�C 2ˇ

�=�Z r

r.t/�0� r

r.t/� 1

�@r Qu Qu

�

CP2

�3=�Z

�� r

r.t/� 1

�@r Qu Qu

�Cˇs �P2

�3=�Z

�� r

r.t/� 1

�@r Qu Qu

�

Cˇ.b �P1/

�3=�Z

�� r

r.t/� 1

�@r Qu Qu

�

�b

�3

��s�C b �P1

�=�Z

�� r

r.t/� 1

�@r Qu Qu

�

Cˇ

�<�Z

i@t Quh� 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�QuC 2�

� r

r.t/� 1

�@r Qu

i�

Dˇ

�<�Z

i@t Quh� 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�QuC 2�

� r

r.t/� 1

�@r Qu

i�

CO� b�4k"k2

H1�

�; (5.12)


where we used in the last inequality (4.35) the bootstrap assumptions (4.21), (4.22),(4.23), and (4.24), and the fact that

1

r�

1

r.t/on the support of �

� �r.t/� 1

�: (5.13)

The first term in the right-hand side of (5.12) corresponds to the localizedMorawetz multiplier, and we get, from (5.1) and the classical Pohozaev integration-by-parts formula,

ˇ

�<�Z

i@t Quh� 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�QuC 2�

� r

r.t/� 1

�@r Qu

i�

D˛b

�2

�Z�0� r

r.t/� 1

�j@r Quj

2�

�˛2b2

8ˇ�3

�Z�� 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�j Quj2

�

�2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�

�ˇ

�<�Z � 1

r.t/�0C

N � 1

r�� r

r.t/� 1


�Qu�

�2ˇ

�<�Z

�� r

r.t/� 1

� @r Qu

�

�ˇ

�<�Z � 1

r.t/�0C

N � 1

r�� r

r.t/� 1

� Qu�;


ˇ

�<�Z

i@t Quh� 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�QuC 2�

� r

r.t/� 1

�@r Qu

i�

D�2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�

�ˇ

�<�Z � 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�

��f . QQC Qu/� f . QQ/� f 0. QQ/ � Qu

�Qu�

�2ˇ

�<�Z

�� r

r.t/� 1

� @r Qu

�

�ˇ

�<�Z � 1

r.t/�0C

N � 1

r�� r

r.t/� 1

� Qu�CO

� b�4k"k2

H1�

�: (5.14)


We estimate by direct inspection,ˇ̌f .1C z/� f .1/� f 0.1/ � z

ˇ̌� jzjp C jzj21p>2;

and hence the bound by homogeneity,ˇ̌f . QQC Qu/� f . QQ/� f 0. QQ/ � Qu

ˇ̌� j Qujp C j QQjp�2j Quj21p>2: (5.15)

We thus obtain the boundˇ̌̌�ˇ

�<�Z � 1

r.t/�0C

N � 1

r�� r

r.t/� 1

��f . QQC Qu/� f . QQ/� f 0. QQ/ � Qu

�Qu�ˇ̌̌

� b

�2

hZj QujpC1C j Quj3j QQjp�21p>2

i; (5.16)

where we used (5.13). We claim the nonlinear bounds

Zj Quj3j QQjp�2 �

ık"k2L2�

�2for p > 2; (5.17)

Zj QujpC1 �

ıp�1k"k2H1�

�2; (5.18)

which are proved below. The terms involving in (5.14) are estimateed in brute forceusing (5.10):

ˇ̌̌2ˇ�<�Z

�� r

r.t/� 1

� @r Qu

�ˇ̌̌

Cˇ̌̌ˇ�<�Z � 1

r.t/�0C

N � 1

r�� r

r.t/� 1

� Qu�ˇ̌̌

�.bk C bk"kH1�/k"kH1�

�4: (5.19)

Injecting (5.16), (5.17), (5.18), and (5.19) into (5.14) yields

ˇ

�<�Z

i@t Quh� 1

r.t/�0C

N � 1

r�� r

r.t/� 1

�QuC 2�

� r

r.t/� 1

�@r Qu

i�

D�2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�

CO� b�4k"k2

H1�Cbk

�4k"kH1�

�:

We now inject this into (5.12), which together with (5.11) concludes the proof of (5.6).


Proof of (5.17)Note first that QQ is localized in the region r � r.t/=2 due to the cutoff �b in itsdefinition. Now, the region r � r.t/=2 corresponds to y �� r.t/

2�.t/and thus

�� 1 for r � r.t/=2:

For p > 2, we estimate from the Sobolev bound (4.43) and the bootstrap assumption(4.21), Z

j Quj3j QQjp�2 D1

�2

Zy�� r.t/

2�.t/

j"j3jQb; Q̌jp�2�� 1

�2

Zy�� r.t/

2�.t/

j"j3�

�1

�2k"k

L1.y�� r.t/2�.t/

/k"k2

L2�� 1

�2k"kH1�k"k

2

L2�

�ık"k2

H1�

�2;

and (5.17) is proved.

Proof of (5.18)Observe that the bootstrap bound (4.21) implies

k QukH1 �k"kH1�

�� ı: (5.20)

In view of the Sobolev embeddings, this yields

Zj QujpC1 � k QukpC1

H1�k"k2

H1�

�2k Quk

p�1

H1�ıp�1k"k2

H1�

�2;

and (5.18) is proved. This concludes the poof of Lemma 5.1.

5.2. Coercivity of I

We now examine the various terms in Lemma 5.1 which correspond to quadraticinteractions. Let us start with the boundary term in time I.

LEMMA 5.2 (Coercivity of I)Let I. Qu/ given by (5.4). Then

I. Qu/� c0

�kr Quk2

L2C

1

�2k Quk2

L2

�(5.21)

for some universal constant c0 > 0.


Proof of Lemma 5.2We first renormalize

I. Qu/D1

2�2

°Zj@y"j

2�C 2ˇ=�Z

�.z/@y""��C .1C ˇ2/

Zj"j2�

� 2

Z �F.Q

b; Q̌C "/�F.Q

b; Q̌/�F 0.Q

b; Q̌/ � "

��±;

where

z Dr

r.t/� 1D

˛b

2ˇy; �D .1C z/N�1: (5.22)

We compute

F 00.Qb; Q̌/ � " � "D

p � 1

4Q2

b; Q̌"2C

pC 1

2jQ

b; Q̌jp�1j"j2C

p � 1

4jQ

b; Q̌jp�3Q2

b; Q̌"2

and estimate by homogeneityˇ̌̌F.QC "/�F.Q

b; Q̌/�F 0.Q

b; Q̌/ � "�

1

2F 00.Q

b; Q̌/ � " � "

ˇ̌̌� j"jpC1C j"j3jQ

b; Q̌jp�21p>2: (5.23)

We conclude using the bounds (5.17) and (5.18),

2

Z �F.Q

b; Q̌C "/�F.Q

b; Q̌/�F 0.Q

b; Q̌/ � "

��

D

Z hp � 14

Q2

b; Q̌"2C

pC 1

2jQ

b; Q̌jp�1j"j2C

p � 1

4jQ

b; Q̌jp�3Q2

b; Q̌"2i�

CO��2Zj Quj3j QQjp�21p>2C j QujpC1

�

D p

ZQ"21�bQ

p�1C

ZQ"22�bQ

p�1CO�bk"k2

L2�Ck"k3

H1�

�D p

ZQ"21�bQ

p�1C

ZQ"22�bQ

p�1CO�ıC k"k2

H1�

�;

where we used the estimates (5.17) and (5.18), the bootstrap assumption (4.22), thefact that

Qb; Q̌D �bQe

�iˇyCO.be�cjyj/ and �bQp�1�D �bQ

p�1CO.b�be�cjyj/;

and where we recall from (4.20) that

Q"D "eiˇy :


Together with ˇ D ˇ1 C Q̌ and the bootstrap assumptions (4.22) and (4.23), thisyields the preliminary estimate

I. Qu/D1

2�2

°Zj@y"j

2�C 2ˇ1=�Z

�.z/@y""��C

Z.1C ˇ21/j"j

2�

� p

ZQ"21Q

p�1 �

ZQ"22Q

p�1CO�ıC k"k2

H1�

�±: (5.24)

Let us now split the potential part in the zones jyj � 1pb

, jyj � 1pb

. Away from thesoliton, the reduced discriminant of the quadratic form

j@y"j2C 2ˇ1=

��.z/@y""

�C .1C ˇ21/j"j

2

is given by

�D ˇ21�2.z/� .1C ˇ21/

2 < 0

from (5.3), and thusZjyj� 1p

b

�j@y"j

2C 2ˇ1=��.z/@y""

�C .1C ˇ21/j"j

2��Zjyj� 1p

b

�j@y"j

2C j"j2�:

On the singularity jyj� 1pb

, we have from (5.3)

ˇ̌�.z/� 1

ˇ̌� jzj�

pb;

and thus Zjyj� 1p

b

�j@y"j

2C 2ˇ1=��.z/@y""

�C .1C ˇ21/j"j

2��

D

Zjyj� 1p

b

�j@y Q"j

2C jQ"j2�CO

�pbk"k2

H1�

�:

Collecting the above bounds yields

2I. Qu/D

Zjyj� 1p

b

�j@y Q"j

2C jQ"j2�� p

ZQ"21�bQ

p�1C

ZQ"22�bQ

p�1

C

Zjyj� 1p

b

�j@y"j

2C j"j2��CO

�ıC k"k2

H1�

�: (5.25)

We now recall the following coercivity property of the linearized energy in the one-dimensional subcritical case which is a well-known consequence of the variationalcharacterization of Q (see, e.g., [6]).


LEMMA 5.3 (Coercivity of the linearized energy)There holds for some universal constant c0 > 0, 8" 2H 1.R/,�

LC."1/; "1�C�L�."2/; "2

�� c0k"k

2H1�1

c0

®."1;Q/

2C ."1; yQ/2C ."2;ƒQ/

2¯: (5.26)

We now inject the choice of orthogonality conditions (4.17) into (5.26) and obtain,using a standard localization argument,�Z

jyj� 1pb

�j@y Q"j

2C jQ"j2�� p

ZQ"21�bQ

p�1C

ZQ"22�bQ

p�1

�Zjyj� 1p

b

�j@y Q"j

2C jQ"j2�CO

�ıC kQ"k2

H1�

�

�Zjyj� 1p

b

�j@y"j

2C j"j2��CO

�ıC k"k2

H1�

�;

which together with (5.25) concludes the proof of (5.21).

Remark 5.4One can easily extract from the above proof the upper bound

I � kr Quk2L2C

1

�2k Quk2

L2: (5.27)

5.3. Estimate for J. Qu/

We now treat the J. Qu/ term given by (5.5). We first extract the leading order quadraticterms in J. Qu/ and claim that is a b degenerate quadratic term. A suitable choice ofthe cutoff function � would allow us sign this term again as in [38], but we shall notneed this additional structural fact here.

LEMMA 5.5 (Leading order terms in J. Qu/)We have the rough bound

ˇ̌J. Qu/

ˇ̌� b

�4k"k2

H1�: (5.28)

Proof of Lemma 5.5Step 1: The @t QQ term. We compute @t QQ from (3.34):

�Using the smallness of b and the exponential localization of Q; see, for example, [19].


QQt D i�t QQ�2

p � 1

�t

�QQ�

r � r.t/

�

�t

�

1

�2p�1

Q0b; Q̌

�r � r.t/�.t/

�ei�

�rt .t/

�

1

�2p�1

Q0b; Q̌

�r � r.t/�.t/

�ei� C bt

1

�2p�1

@bQb.t/; Q̌.t/

�r � r.t/�.t/

�ei�.t/

C Q̌t1

�2p�1

@ Q̌Qb.t/; Q̌.t/

�r � r.t/�.t/

�ei�.t/

D� i.1C ˇ2/

�2C

2

p � 1

b

�2

�QQC

b

�

r � r.t/

�@r QQC

2ˇ

�@r QQ

C1

�2C2p�1

O�hb2CMod C

ˇ̌̌bs C .1� ˛/b

2 �b

ˇP2 � bP1

ˇ̌̌i�bjyj

ce�jyj�

Di.1C ˇ2/

�2QQC

2ˇ

�@r QQC

1

�2C2p�1

O�b�bjyj

ce�jyj�;

where we used (4.35) and the decay estimate (3.15) in the last step. This yields

�<�@t QQ;

�f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

��D�

1C ˇ2

�2=�Z �

f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu�QQ�

�2ˇ

�<�Z �

f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu�@r QQ

�

C1

�4O�Z

b�bjyjce�jyj

ˇ̌f .Q

b; Q̌C "/� f .Q

b; Q̌/� f 0.Q

b; Q̌/ � "

ˇ̌��;

where we used the estimates (5.17) and the bootstrap assumptions (4.21) and (4.22).We estimate the nonlinear terms using (4.45), (4.21), and (4.43):Z

b�bjyjce�jyj

ˇ̌f .Q

b; Q̌C "/� f .Q

b; Q̌/� f 0.Q

b; Q̌/ � "

ˇ̌�

� bZ�bjyj

ce�jyj�jQ


��

� b�1Ck"k

p�2

L1.y�� ıb/1p>2

� Zj"j2�� bk"k2

H1�:

Injecting the collection of above bounds into (5.5) yields the preliminary computation

J. Qu/D�1C ˇ2

�2=

Z �f .u/� f . QQ/; Qu

�

�2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�


�1C ˇ2

�2=�Z �


�2ˇ

�<�Z �


�

CO� b�4k"k2

H1�

�: (5.29)

Step 2: Nonlinear cancellation on the phase term. We observe using the explicitformula for f and

f 0. QQ/ � QuDpC 1

2j QQjp�1 QuC

p � 1

2j QQjp�3 QQ2 Qu (5.30)

the nonlinear cancellation

�1C ˇ2

�2=

Z �f .u/� f . QQ/; Qu

�

�1C ˇ2

�2=�Z �


D�1C ˇ2

�2=�Z

f . QuC QQ/; QQC Qu�C1C ˇ2

�2=�Z

f . QQ/; QQ�

C1C ˇ2

�2=�Z

f . QQ/ QuC f 0. QQ/ � Qu QQ�

D1C ˇ2

�2=�Z

f . QQ/ QuC f 0. QQ/ � Qu QQ�

D1C ˇ2

�2=�Zj QQjp�1

�QQ QuC

pC 1

2Qu QQC

p � 1

2QQ Qu��

D 0: (5.31)

Step 3: Conclusion. Let ' be a smooth compactly supported cutoff function whichis 1 in the neighborhood of the support of �, and 0 in the neighborhood of z D �1.We compute

A1 D �2ˇ

�<�Z

'� r

r.t/� 1


�@r Qu

�

�2ˇ

�<�Z

'� r

r.t/� 1

��f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

�@r QQ

�

D �2ˇ

�<�Z

'� r

r.t/� 1

�f . QQC Qu/@r QQC @r Qu

�

C2ˇ

�<�Z

'� r

r.t/� 1

�f . QQ/@r QQ

�


C2ˇ

�<�Z

'� r

r.t/� 1

��f . QQ/@r QuC f

0. QQ/ � Qu@r QQ��

D �2ˇ

�<

Z'� r

r.t/� 1

�@r�F.u/�F. QQ/� f . QQ/ Qu

�:

Integrating by parts in r , we obtain

A1 D2ˇ

�<

Z h 1

r.t/'0C

N � 1

r'i� r

r.t/� 1

��F.u/�F. QQ/� f . QQ/ Qu

�:

In view of the properties of ', we have

1

r�

1

r.t/on the support of '

� �r.t/� 1

�; (5.32)

and thus

A1 D2ˇ

�<

Z h 1

r.t/'0C

N � 1

r'i� r

r.t/� 1

�(5.33)

��F.u/� F. QQ/� f . QQ/ Qu�

1

2F 00. QQ/. Qu; Qu/

�CO

� b�4k"k2

H1�

�:

Next, we estimate using (5.23), the nonlinear estimates (5.17) and (5.18), (5.32), and(4.15),

ˇ̌̌2ˇ�<

Z h 1

r.t/'0C

N � 1

r'i� r

r.t/� 1

�

��F.u/�F. QQ/� f . QQ/ Qu�

1

2F 00. QQ/. Qu; Qu/

�ˇ̌̌

� b

�4

Z �j"jpC1C j"j3jQ

b; Q̌jp�21p>2

�� bıC

�4k"k2

H1�;


A1 DO� b�4k"k2

H1�

�: (5.34)

Since ' D 1 on the support of �, we have

2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�

D2ˇ

�<�Z

.'�/� r

r.t/� 1


�@r Qu

�;

and thus from (4.15),


A2 D�2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�

C2ˇ

�<�Z

'� r

r.t/� 1


�@r Qu

�

D�2ˇ

�<�Z

'� r

r.t/� 1

�h�� r

r.t/� 1

�� 1

i


�@r Qu

�: (5.35)

We then observe the identity

<�@r�F. QQC Qu/�F. QQ/� f . QQ/ Qu

��f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

�@r QQ

�D<

�f . QQC Qu/.@r QQC @r Qu/� f . QQ/@r QQ� f

0. QQ/ � @r QQ Qu� f . QQ/@r Qu

� f . QQC Qu/@r QQC f . QQ/@r QQC f0. QQ/ � Qu@r QQ

�D<


�@r Qu

�C<

��f 0. QQ/ � @r QQ QuC f

0. QQ/ � Qu@r QQ�

D<��f . QQC Qu/� f . QQ/

�@r Qu

�C<

��@uf . QQ/@r QQ Qu� @uf . QQ/@r QQ QuC @uf . QQ/ Qu@r QQC @uf . QQ/ Qu@r QQ

�D<


�@r Qu

�C<

��@uf . QQ/@r QQ QuC @uf . QQ/ Qu@r QQ

�D<


�@r Qu

�;

where we used in the last inequality the fact that

@uf . QQ/DpC 1

2j QQjp�1 2R:

Injecting this into (5.35) and using (4.15), (5.22) yields

A2 D �2ˇ

�<�Z

'� r

r.t/� 1

�h�� r

r.t/� 1

�� 1

i

� @r�F. QuC QQ/�F. QQ/� f . QQ/ Qu

��

C2ˇ

�<�Z

'� r

r.t/� 1

�h�� r

r.t/� 1

�� 1

i

��f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

�@r QQ

�

D2ˇ

�2<�Z �˛b

2ˇ@z�'.z/

��.z/� 1

��C.N � 1/�

r'.z/

��.z/� 1

�� r

r.t/� 1

�

��F. QuC QQ/�F. QQ/� f . QQ/ Qu

��


C2ˇ

�<�Z

'� r

r.t/� 1

�h�� r

r.t/� 1

�� 1

i

��f . QuC QQ/� f . QQ/� f 0. QQ/ � Qu

�@r QQ

�: (5.36)

Since �.0/D 1, we obtain from (5.22)ˇ̌�.z/� 1

ˇ̌� jzj� bjyj;

ˇ̌@z��.z/� 1

�ˇ̌� 1: (5.37)

We inject this into (5.36) and use the homogeneity bounds (4.44) andˇ̌F.Q

b; Q̌C "/�F.Q

b; Q̌/� f .Q

b; Q̌/"ˇ̌� jQ

b; Q̌jp�1j"j2C j"jpC1;

the pointwise bound

ˇ̌�.z/

ˇ̌Cˇ̌@z�.z/

ˇ̌� 1; �

r

�1C jzj

��

r

ˇ̌̌ rr.t/

ˇ̌̌� b on Supp.�/;

and the decay (3.15) to estimate

A2 �b

�4

Z �jQ

b; Q̌jp�1j"j2C j"jpC1

��ˇ̌@z.'.z/

��.z/� 1

�ˇ̌C '.z/

ˇ̌�.z/� 1

ˇ̌��

C1

�4

Z �jQ


�'.z/

ˇ̌�.z/� 1

ˇ̌j@yQb; Q̌

j�

� b

�4

Z �j"jpC1C bjyjC �be

�.p�1/jyjj"j2��

C1

�4

Zbjyj

�j"j2jyjC �be

�.p�1/jyjC �bj"jp1p>2e�cjyj

��

� b

�4k"k2

H1�; (5.38)

where we used (5.18) and the Sobolev bound (4.43) in the last step. We conclude from(5.34) and (5.38) that

A1 �A2 DO� b�4k"k2

H1�

�: (5.39)

The function 1 � ' is supported by construction in y � � 1b

where QQ vanishes, andhence (5.39) ensures

�2ˇ

�<�Z

�� r

r.t/� 1


�@r Qu

�

�2ˇ

�<�Z �


�DO

� b�4k"k2

H1�

�: (5.40)


In view of (5.29), (5.31), and (5.40), we obtain the expansion of quadratic termsin J. Qu/,

J. Qu/DO� b�4k"k2

H1�

�;

which is the wanted estimate (5.28). This concludes the proof of Lemma 5.5.

6. Existence of ring solutionsWe conclude in this section the proof of Theorem 1.2. We start with closing the boot-strap Proposition 4.3 using the monotonicity tools developed in the previous section,and we then prove the existence of a ring solution using a now-standard Schauder-typecompactness argument and a backward integration of the flow from blow-up time.

6.1. Closing the bootstrapWe are now in position to close the bootstrap, that is, Proposition 4.3.

Proof of Proposition 4.3Step 1: Pointwise control of ". In view of (5.6), we have

d.I. Qu/��/

dtD

1

��dI. Qu/

dt�

�t

��C1I. Qu/

D1

��J. Qu/C

b

�2C�I. Qu/�

P1

�2C�I. Qu/

��s�C b �P1

� I.t/

�2C�CO

� b

�4C�k"k2

H1�C

bk

�4C�k"kH1�

�:

We estimate from (4.35), the bootstrap assumptions (4.22) and (4.23), and (5.27) thatˇ̌̌ P1

�2C�I. Qu/

ˇ̌̌Cˇ̌̌��s�C b

� I.t/

�2C�

ˇ̌̌� b

�4C�

�b2C bk"kH1� C b

k�k"k2

H1�

� bıC

�4C�k"k2

H1�;

which together with (5.21) yields the ecistence of a constant C > 0 such that

d.I. Qu/��/

dt� .c0 �C/

b

�4C�k"k2

H1��C

b2k�1

�4C�: (6.1)

We fix such that

>C

c0:

Then, (6.1) yields

d.I. Qu/��/

dt��b

2k�1

�4C�: (6.2)


Now, the definition (4.15) of b together with the bootstrap assumptions (4.22), (4.23),and (4.24) yield

b � �1�˛: (6.3)

In view of (6.2) and (6.3), we obtain

d.I. Qu/��/

dt��b�2.1�˛/.k�1/�4�� : (6.4)

This yields, after integration between t and Nt using I. Qu/D 0 at t D Nt from the well-prepared initial data assumption (4.8),

I. Qu/� �.t/�Z Ntt

b.�/�.�/2.1�˛/.k�1/�4�� d�: (6.5)

We now use the bootstrap bounds (4.21), (4.22), and (4.23) and the modulation equa-tion (4.35) to estimateˇ̌̌�s

�C b

ˇ̌̌�ˇ̌P1.b; Q̌/

ˇ̌C bk"kH1� C b

k � ıCb from which 0 < b ��t :

We conclude from (6.5) and the choice of k,

k > 1C1Cmax. �

2; 1/

1� ˛;

that

I. Qu/� �.t/2:

In view of the coercivity (5.21) of I. Qu/, we obtain

kr Quk2L2C

1

�2k Quk2

L2� �2

or, equivalently,

k"kH1� � �2: (6.6)

Step 2: Control of the modulation parameters. To conclude the proof of Proposi-tion 4.3, it remains to control the modulation parameters. We first derive an estimatefor Mod.t/. Note that in view of (6.4), we obtain the following improvement of (6.6):

k"kH1� � �2C.1�˛/.k�1� 21�˛ /:

Together with (4.35) and (6.3), we deduce

Mod.t/� bk"kH1� C bk � bk : (6.7)

The control of the modulation parameters is achieved by the following lemma.


LEMMA 6.1Let k satisfy the condition

k >2

1� ˛C 1: (6.8)

Let te be as defined in Lemma 4.1. Let t and Nt such that te � t < Nt < 0. Let .�e; be; Q̌e;re; �e/ be the solution to the exact system (4.1) of modulation equations. Let .�; b; Q̌;r; �/ be initialized at t D Nt as

.�; b; Q̌; r; �/.Nt /D .�e; be; Q̌e; re; �e/.Nt / (6.9)

and be the solution of the following perturbed system of modulation equations onŒt ; Nt �: 8̂̂

ˆ̂̂̂̂<ˆ̂̂̂̂ˆ̂:

�s�C b �P1.b; Q̌/DO.b

k/;rs�C 2ˇDO.bk/;

Q̌s �P2.b; Q̌/DO.b

k/;


�s D 1C ˇ2CO.bk/:

(6.10)

Then, the following bounds hold on Œt ; Nt �:

b.t/D1

1C ˛

�2.1C ˛/ˇ1˛g1

� 21C˛jt j

1�˛1C˛

�1CO


1�˛1C˛

��; (6.11)

�.t/D�2.1C ˛/ˇ1

˛g1

� 11C˛jt j

11C˛

�1CO


1�˛1C˛

��; (6.12)

r.t/D g1

�2.1C ˛/ˇ1˛g1

� ˛1C˛jt j

˛1C˛

�1CO


1�˛1C˛

��; (6.13)

Q̌.t/D O�jt j

2.1�˛/1C˛

�; (6.14)

and

�.t/D .1C ˇ21/� 1� ˛1C ˛

� 1�˛1C˛

�2.1� ˛/ˇ1˛g1

�� 21C˛jt j�

1�˛1C˛ CO

�log�jt j��: (6.15)

The proof of Lemma 6.1 is postponed to Appendix B. We now conclude theproof of Proposition 4.3. The assumptions (6.8), (6.9), and (6.10) of Lemma 6.1 aresatisfied in view of the choice (4.7) for k, (6.7), and (4.8). Thus, the conclusions ofLemma 6.1 apply. In particular, (6.11) yields (4.26), (6.14) yields (4.27), (6.13) and(6.12) yield (4.28), while (6.6) and (6.12) yield (4.25). This concludes the proof ofProposition 4.3.


6.2. Proof of Theorem 1.2We are now in position to conclude the proof of Theorem 1.2.

Proof of Theorem 1.2Let .tn/n�1 be an increasing sequence of times tn < 0 such that tn! 0�. Let un bethe solution to (1.1) with initial data at t D tn given by

un.tn; r/D1

�e.tn/2p�1

Qbe.tn/; Q̌e.tn/

�r � re.tn/�e.tn/

�ei�e.tn/: (6.16)

Let t < 0 be the backward time provided by Proposition 4.3 which is independentof n. We first claim that un.t/ is compact in L2 as n!C1. Indeed, Proposition 4.3ensures the uniform bound

8t 2 Œt ; tn�;��un.t/��H1 � 1: (6.17)

This shows that, up to a subsequence, .un.t//n�1 is compact in L2.r < R/ as n!C1 for allR > 0. TheL2 compactness of un.t/ is now the consequence of a standardlocalization procedure. Indeed, let a cut-off unction .x/D 0 for jxj � 1 and .x/D1 for jxj � 2; then

ˇ̌̌ ddt

ZRjunj

2ˇ̌̌D 2

ˇ̌̌=�ZrR � run

�un/

ˇ̌̌� 1

R;

where we used (6.17). Integrating this backward from tn to t and using (6.16) yields

limR!C1

supn�1

��un.t/��L2.r>R/ D 0;which together with the L2.r < R/ compactness of .un.t//n�1 provided by (6.17)implies, up to a subsequence,

un.t/! u.t/ in L2 as n!C1:

Let then u 2 C.Œt ; T /;H 1/ be the solution to (1.1) with initial data u.t/; then,using the uniform control in H 1 for un and the convergence in L2 of un.t/, weobtain 8t 2 Œt ;min.T; 0//,

un.t/! u.t/ in L2:

Let .�n.t/; bn.t/; �n.t/; "n.t// be the geometrical decomposition associated to un.t/,

un D1

�n.t/2p�1

.Qbn.t/C "n/�t;r � rn.t/

�n.t/

�ei�n.t/I


then u admits on Œt ;min.T; 0// a geometrical decomposition of the form

uD1

�.t/2p�1

.Qb.t/; Q̌.t/

C "/�t;r � r.t/

�.t/

�ei�.t/

with, 8t 2 Œt ;min.T; 0//,

�n.t/! �.t/; rn.t/! r.t/; bn.t/! b.t/; Q̌n.t/! Q̌.t/;

�n.t/! �.t/; and "n.t/! ".t/ in L2 as n!C1

(see [24] for related statements). By passing to the limit in the bounds provided byProposition 4.3 and Lemma 6.1, we obtain the bounds, 8t 2 Œt ;min.T; 0//,

b.t/D1

1C ˛

�2.1C ˛/ˇ1˛g1

� 21C˛jt j

1�˛1C˛

�1CO


1�˛1C˛

��;

�.t/D�2.1C ˛/ˇ1

˛g1

� 11C˛jt j

11C˛

�1CO


1�˛1C˛

��;

r.t/D g1

�2.1C ˛/ˇ1˛g1

� ˛1C˛jt j

˛1C˛

�1CO


1�˛1C˛

��;

Q̌.t/DO�jt j

2.1�˛/1C˛

�;

�.t/D .1C ˇ21/� 1� ˛1C ˛

� 1�˛1C˛

�2.1� ˛/ˇ1˛g1

�� 21C˛jt j�

1�˛1C˛ CO

�log�jt j��;

and

k"kH1� � jt j 21C˛ :

This yields that u 2 C.Œt ; 0/;H 1/, and u blows up at time T D 0. The estimates(1.13), (1.14), and (1.15) are now a straightforward consequence of the above esti-mates for .b;�; r; �; "/.

This concludes the proof of Theorem 1.2.

Appendix A. Integration of the exact system of modulation equationsThe goal of this appendix is to prove Lemma 4.1. For convenience, we prove Lem-ma 4.1 in time s, with ds

dtD 1

�2e.t/. This is done in the following lemma.

LEMMA A.1 (Integration of the exact system of modulation equations in time s)There exists a universal constant s0 1 such that the following holds. Let

1

2< g0 < 1; �0 2R


and

b0 D1

.1� ˛/s0: (A.1)

Then the solution .�e; be; Q̌e; re; �e/ to the dynamical system

.M1/D

8̂̂ˆ̂̂̂̂<ˆ̂̂̂̂ˆ̂:

�s�C b DP1.b; Q̌/;

rs�C 2ˇD 0;

Q̌s DP2.b; Q̌/;


�s D 1C ˇ2

with

8̂̂ˆ̂̂<ˆ̂̂̂̂:

r.s0/�˛.s0/

D g0;

b.s0/D b0;

Q̌.s0/D1

s20

;

�.s0/D �0;

(A.2)

is defined on Œs0;C1/. Moreover, there exists g1 > 0 with

g1 D g0C os0!C1.1/ (A.3)

such that the following asymptotics hold on Œs0;C1/:

be.s/D1

.1� ˛/sCO

� j log sj

s2

�;

ˇ̌Q̌e.s/

ˇ̌� 1

s2; (A.4)

�e.s/Dh ˛g1

2.1� ˛/ˇ1s

i 11�˛

h1CO

� log.s/

s

�i; (A.5)

re.s/D g1

h ˛g1

2.1� ˛/ˇ1s

i ˛1�˛

h1CO

� log.s/

s

�i; (A.6)

and

�e.s/D .1C ˇ21/sCO.1/: (A.7)

We first show how Lemma A.1 yields the conclusion of Lemma 4.1.

Proof of Lemma 4.1In view of (A.5), we have Z C1

s0

�2 <C1:

Thus, since dsdtD 1

�2.t/, the time of existence of the existence of the dynamical system

in time t is finite, and we may choose the origin of time t such that the final time is 0.Then, for all te � t < 0, we have

�t D

Z C1s

�2;


which together with (A.5) yields

1

sD�1C ˛1� ˛

� 1�˛1C˛

�2.1� ˛/ˇ1˛g1

� 21C˛jt j

1�˛1C˛

�1CO


1�˛1C˛

��: (A.8)

Injecting (A.8) in (A.4), (A.5), and (A.7) yields the wanted estimates (4.2), (4.3),(4.4), (4.5), and (4.6). This concludes the proof of Lemma 4.1.

We now turn to the proof of Lemma A.1.

Proof of Lemma A.1Step 1: Reformulation and bootstrap bounds. The local existence of solutions to (A.2)follows from Cauchy–Lipschitz. To control the solution on large positive times, let usintroduce the auxiliary function

gDr

�˛;

which from (A.2) satisfies

dg

dsD�˛

��s�C b

� r�˛D�˛P1.b; Q̌/g: (A.9)

We view equivalently (A.2) as a system on .b; g; Q̌/ with, from direct computation,the equivalent system of equations8̂̂

ˆ̂<ˆ̂̂̂:

dgdsD�˛P1.b; Q̌/g;

bs C .1� ˛/b2 D b

ˇP2.b; Q̌/C bP1.b; Q̌/;

Q̌s DP2.b; Q̌/;


with

8̂̂<ˆ̂:g.s0/D g0;

b.s0/D b0;

Q̌.s0/D1

s20

:

(A.10)

We bootstrap the following a priori bounds on the solution which are consistent withthe initial data:

8s0 � s � Ns;ˇ̌g.s/

ˇ̌� 1C 2g0;

ˇ̌Q̌.s/

ˇ̌�1

s32

;

(A.11)ˇ̌̌b.s/�

1

.1� ˛/s

ˇ̌̌�.log s/2

s2:

Step 2: Closing the bootstrap. We claim that the bounds (A.11) can be improvedon Œs0; Ns � provided s0 has been chosen large enough. Indeed, let us close the b bound.From (A.10) and (A.11),

ˇ̌̌�d

ds

�1b

�C 1� ˛

ˇ̌̌�j bˇ

P2.b; Q̌/C bP1.b; Q̌/j

b2� b.b2C j Q̌j2/

b2� 1

s;


and thus using the boundary condition on b at s0 and the initialization (A.1) we have

ˇ̌̌ 1

b.s/� .1� ˛/s

ˇ̌̌�ˇ̌̌ 1

b.s0/� .1� ˛/s0

ˇ̌̌C

Z s

s0

d�

�� log s;

from which using s � s0 1,ˇ̌̌b.s/�

1

.1� ˛/s

ˇ̌̌� log s

s2: (A.12)

Next, we consider Q̌. Since

P2.b; Q̌/D�2b Q̌ CO.b3C Q̌3/;

we obtain, in view of (A.11),

Q̌s D�2b Q̌ CO

� 1s3

�;

which we rewrite using (A.11) and (A.12):

ˇ̌̌ dds.s

21�˛ Q̌/

ˇ̌̌� s 2

1�˛

h log s

s2j Q̌j C

1

s3

i� s 2

1�˛�3:

We integrate using the boundary condition (A.10) and 21�˛� 2 > 0:

ˇ̌s

21�˛ Q̌.s/

ˇ̌� js

21�˛

0Q̌0j C s

21�˛�2 � s

21�˛�2

0 � s 21�˛�2;

and thus

ˇ̌Q̌.s/

ˇ̌� 1

s2: (A.13)

We now close the g bound in brute force from (A.12), (A.13), and (A.10), which yield

ˇ̌̌dgds

ˇ̌̌� 1C 2g0

s3;

and thus, in view of the initialization (A.10),

ˇ̌g.s/

ˇ̌� g0CC

1C g0

s20�1

2C3

2g0; (A.14)

for s0 large enough. The bounds (A.12), (A.13), and (A.14) improve (A.11), and thusfrom a standard continuity argument, the bounds (A.12), (A.13), and (A.14) hold onŒs0;C1/ and the solution is global.

Step 3: Conclusion. The bounds (A.12) and (A.13) being now global, (A.4) isproved. We moreover conclude from (A.10) that


Z C1s0

ˇ̌̌dgds

ˇ̌̌ds �

Z C1s0

ds

s3D os0!C1.1/;

and hence there exists g1 satisfying (A.3) such that

8s � s0;ˇ̌g.s/� g1

ˇ̌� 1

s2: (A.15)

This yields from (A.10), (A.15), and (A.11)

�.s/D˛b

2ˇr D

˛

2.1� ˛/ˇ1sg1�

˛h1CO

� log.s/

s

�i;

from which

�.s/Dh ˛g1

2.1� ˛/ˇ1s

i 11�˛

h1CO

� log.s/

s

�i:

Together with (A.15), we obtain

r.s/D g�˛ D g1

h ˛g1

2.1� ˛/ˇ1s

i ˛1�˛

h1CO

� log.s/

s

�i:

Finally, it only remains to estimate � . In view of (A.2) and (A.13), we have

d�

dsD 1C ˇ21CO

� 1s2

�;

which after integration between s0 and s yields

�.s/D .1C ˇ21/sCO.1/:

This concludes the proof of Lemma A.1.

Appendix B. Stability of the modulation equationsThe goal of this Appendix is to prove Lemma 6.1. Recall that te is defined in Lem-ma 4.1 and that t and Nt are such that te � t < Nt < 0. We introduce the auxiliaryfunctions

gDr

�˛and ge D

re

�˛e:

We define

b D b � be; gD g � ge; and Q̌ D Q̌ � Q̌e:

From the initial conditions (6.9), we have

.b; g; Q̌/.Nt /D .0; 0; 0/: (B.1)


Let t0 such that t � t0 < Nt and such that the following bootstrap assumptions� hold:

8t0 � t � Nt ;ˇ̌g.t/

ˇ̌Cˇ̌Q̌.t/

ˇ̌Cˇ̌b.t/

ˇ̌� jt j

21C˛ : (B.2)

Next, recall (A.2):

dge

dsD�˛P1.be; Q̌e/ge: (B.3)

Also, the modulation equations for r and � and the choice of b in (6.10) implies

dg

dsD�rs�C 2ˇ

�g�

r� ˛g

��s�C b

�D�˛P1.b; Q̌/gCO.b

k/: (B.4)

We have

ds

dtD

1

�2D� 2ˇ˛bg

� 21�˛

anddse

dtD

1

�2eD� 2ˇe

˛bege

� 21�˛

: (B.5)

In view of the dynamical system (6.10) for .b; Q̌/, the dynamical system (B.4) for g,the dynamical system (A.2) for .be; ge; Q̌e/, the dynamical system (B.3) for ge , thedefinition of .b; g; Q̌/, (B.5), and the bootstrap bound (B.2), we obtain the following

dynamical system for .b; g; Q̌/ on t0 � t � Nt :

bt �2˛

1C ˛

b

jt jDO

�jt j�

2˛1C˛

�jbj C j Q̌j C jgj

�C jt jk

1�˛1C˛�

21C˛

�; (B.6)

Q̌tC

2

1C ˛

Q̌

jt jDO

�jt j�

2˛1C˛


�C jt jk

1�˛1C˛�

21C˛

�; (B.7)

and

gtDO

�jt j�

2˛1C˛


�C jt jk

1�˛1C˛�

21C˛

�: (B.8)

Integrating (B.8) between t and Nt and using (B.1) and (6.8) yields

ˇ̌g.t/

ˇ̌�Z Ntt

j� j�2˛1C˛


�d� C jt jk

1�˛1C˛C1�

21C˛ : (B.9)


ˇ̌b.t/

ˇ̌� jt j� 2˛

1C˛

Z Ntt


�d� C jt jk

1�˛1C˛C1�

21C˛ : (B.10)


�Which are consistent with (B.1).


ˇ̌Q̌.t/

ˇ̌� jt j 2

1C˛

Z Ntt

j� j�2�jbj C j Q̌j C jgj

�d� C jt jk

1�˛1C˛C1�

21C˛ : (B.11)

In view of (B.9), (B.10), and (B.11), we obtain

jbj C j Q̌j C jgj �Z Ntt

�j� j�

2˛1C˛ C jt j�

2˛1C˛ C jt j

21C˛ j� j�2

��jbj C j Q̌j C jgj

�d�

C jt jk1�˛1C˛C1�

21C˛ : (B.12)

Injecting the bootstrap assumption (B.2), noticing that the integral is convergent, andthen reiterating finally yields

jbj C j Q̌j C jgj� jt jk 1�˛1C˛C1�2

1C˛ ; (B.13)

which is an improvement of the bootstrap assumption (B.2) in view of (6.8). Thus,(B.13) holds on Œt ; Nt �. Now, the wanted estimate (6.11) for b and (6.14) for Q̌ followfrom (B.13) and (6.8), and the estimate (4.2) for be and (4.5) for Q̌e . Also, the wantedestimate (6.12) for � and (6.13) for r follow from the definition of b and g, (B.13)and (6.8), and the estimate (4.4) for re and (4.3) for �e .

Finally, we derive the wanted estimate for � . In view of the dynamical system(6.10) for � , the dynamical system (A.2) for �e , (B.5), and the estimate (B.13), weobtain the following dynamical system for � � �e :

.� � �e/t DO�jt jk

1�˛1C˛C1�

5�˛1C˛

�: (B.14)

Now, recall that �. Nt /D �e. Nt / from (6.9), so that integrating (B.14) between t and Ntand using (6.8) yields

j� � �ej� jt jk1�˛1C˛C2�

5�˛1C˛ : (B.15)

Now, the wanted estimate for � (6.15) follows from (B.15) and (6.8), and the estimatefor �e (4.6). This concludes the proof of Lemma 6.1.

Acknowledgments. The authors thank the anonymous referees for their careful readingof the paper. P.R. would like to thank the MIT Mathematics Department, which he wasvisiting when finishing this work.

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Merle

Laboratoire AGM, Université de Cergy Pontoise, 95302 Cergy-Pontoise, France;

[email protected]

Raphaël

Laboratoire J. A. Dieudonné, Université de Nice Sophia Antipolis, 06 000 Nice, France;

[email protected]

Szeftel

Laboratoire Jacques-Louis Lions, UPMC, 75005, Paris, France; [email protected]


http://dx.doi.org/10.1007/PL00001048


http://dx.doi.org/10.1007/s00208-004-0596-0


http://dx.doi.org/10.1215/S0012-7094-06-13421-X


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http://dx.doi.org/10.1007/s00220-009-0796-2


http://dx.doi.org/10.1090/S0894-0347-2010-00688-1



mailto:[email protected]



on collapsing ring blow-up solutions to the mass supercritical nonlinear schrödinger equation

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