Energy-supercritical NLS: critical H ˙ s -bounds imply scattering
aa r X i v : . [ m a t h . A P ] D ec ENERGY-SUPERCRITICAL NLS:CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING ROWAN KILLIP AND MONICA VISAN
Abstract.
We consider two classes of defocusing energy-supercritical nonlinear Schr¨odingerequations in dimensions d ≥
5. We prove that if the solution u is apriorily bounded inthe critical Sobolev space, that is, u ∈ L ∞ t ˙ H s c x , then u is global and scatters. Introduction
We consider the initial-value problem for the defocusing nonlinear Schr¨odinger equationin dimension d ≥ ( iu t = − ∆ u + F ( u ) u ( t = 0 , x ) = u ( x ) , (1.1)where the nonlinearity F ( u ) = | u | p u is energy-supercritical, that is, p > d − .The class of solutions to (1.1) is left invariant by the scaling u ( t, x ) λ p u ( λ t, λx ) . (1.2)This defines a notion of criticality . More precisely, a quick computation shows that the onlyhomogeneous L x -based Sobolev space left invariant by the scaling is ˙ H s c x ( R d ), where the critical regularity is s c := d − p . If the regularity of the initial data to (1.1) is higher/lowerthan the critical regularity s c , we call the problem subcritical/supercritical .We consider (1.1) for initial data belonging to the critical homogeneous Sobolev space,that is, u ∈ ˙ H s c x ( R d ), in two regimes where s c > d ≥
5. We prove that any maximal-lifespan solution that is uniformly bounded (throughout its lifespan) in ˙ H s c x ( R d ) must beglobal and scatter. We were prompted to consider this problem by a recent preprint ofKenig and Merle [15] which proves similar results for radial solutions to the nonlinear waveequation in R .Let us start by making the notion of a solution more precise. Definition 1.1 (Solution) . A function u : I × R d → C on a non-empty time interval0 ∈ I ⊂ R is a solution (more precisely, a strong ˙ H s c x ( R d ) solution) to (1.1) if it lies in theclass C t ˙ H s c x ( K × R d ) ∩ L p ( d +2) / t,x ( K × R d ) for all compact K ⊂ I , and obeys the Duhamelformula u ( t ) = e it ∆ u (0) − i Z t e i ( t − s )∆ F ( u ( s )) ds (1.3)for all t ∈ I . We refer to the interval I as the lifespan of u . We say that u is a maximal-lifespan solution if the solution cannot be extended to any strictly larger interval. We saythat u is a global solution if I = R .We define the scattering size of a solution to (1.1) on a time interval I by S I ( u ) := Z I Z R d | u ( t, x ) | p ( d +2)2 dx dt. Associated to the notion of solution is a corresponding notion of blowup. As we willsee in Theorem 1.6 below, this precisely corresponds to the impossibility of continuing thesolution.
Definition 1.2 (Blowup) . We say that a solution u to (1.1) blows up forward in time ifthere exists a time t ∈ I such that S [ t , sup I ) ( u ) = ∞ and that u blows up backward in time if there exists a time t ∈ I such that S (inf I,t ] ( u ) = ∞ . We subscribe to the following conjecture.
Conjecture 1.3.
Let d ≥ , p ≥ d , and let u : I × R d → C be a maximal-lifespan solutionto (1.1) such that u ∈ L ∞ t ˙ H s c x . Then u is global and moreover, S R ( u ) ≤ C (cid:0) k u k L ∞ t ˙ H scx (cid:1) , (1.4) for some function C : [0 , ∞ ) → [0 , ∞ ) . Our primary goal in this paper is to demonstrate how techniques developed to treatthe energy-critical NLS can be applied to Conjecture 1.3 in the regime s c ≥
1, althoughsome of the arguments we will use were developed first in the mass-critical setting. As wewill describe, the appearance of the L ∞ t ˙ H s c x norm on the right-hand side of (1.4) rendersillusory the supercriticality of the equation. The famed supercriticality of Navier–Stokes isthe fact that the problem is supercritical with respect to all quantities controlled by (known)conservation/monotonicity laws; see, for instance, the discussion in [31]. In the context ofConjecture 1.3, the assumption that the solution is uniformly bounded in ˙ H s c x plays therole of the missing critical conservation law. It is not surprising therefore that techniquesdeveloped to treat problems with true critical conservation laws should be applicable in thissetting. Next we review some of this work before describing the particular contribution ofthis paper.Mass and energy are the only known coercive conserved quantities for NLS; hence, thecorresponding critical NLS equations have received the most attention. In the mass-criticalcase, the critical regularity is s c = 0 (i.e. p = d ) and the scaling (1.2) leaves the massinvariant, that is, the conserved quantity M ( u ) := Z R d | u ( t, x ) | dx. Similarly, in the energy-critical case, the critical regularity is s c = 1 (i.e. p = d − , d ≥ E ( u ) := Z R d (cid:2) |∇ u ( t, x ) | + p | u ( t, x ) | p (cid:3) dx, which is also a conserved quantity for (1.1).In the defocusing energy-critical case, it is known that all ˙ H x initial data lead to globalsolutions with finite scattering size. Indeed, this was proved by Bourgain [2], Grillakis[11], and Tao [28] for spherically-symmetric initial data, and by Colliander–Keel–Staffilani–Takaoka–Tao [8], Ryckman–Visan [24], and Visan [37, 38] for arbitrary initial data. Forresults in the focusing case see [13, 19].Unlike for the nonlinear wave equation (NLW), all known monotonicity formulae forNLS (that is, Morawetz-type inequalities) scale differently than the energy. Ultimately,the ingenious induction on energy technique of Bourgain (and the concomitant identifica-tion of bubbles) introduces a length scale to the problem which makes it possible to use NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 3 non-invariantly scaling monotonicity formulae. All subsequent work has built upon thisfoundational insight.The Lin–Strauss Morawetz inequality used by Bourgain has ˙ H / x -scaling and is bestadapted to the spherically-symmetric problem. To treat the non-radial problem, Colliander,Keel, Staffilani, Takaoka, and Tao introduced an interaction Morawetz inequality; this has˙ H / x -scaling, which is even further from the ˙ H x -scaling of the energy-critical problem. Asthe reader may notice, in both cases the regularity associated to the monotonicity formulaeis lower than the critical regularity of the equation. This is the philosophical basis ofour belief that the techniques developed for treating the energy-critical problem should bebroadly applicable to Conjecture 1.3 whenever s c ≥ /
2. In this paper, we will by no meanscomplete this program, but rather have chosen to present some selected results that givethe flavor of our main thesis without becoming swamped with technicalities.We turn our attention now to the defocusing mass-critical NLS. In this case, Conjec-ture 1.3 has been proved for spherically-symmetric L x initial data in all dimensions d ≥ L x data with mass less than that of the ground state and d ≥
2) see[18, 21]. At present, we do not know how to deal with the Galilean symmetry possessed bythis equation, except through suppressing it by assuming spherical symmetry. We also notethat in this case, one needs to prove additional regularity (rather than decay) to gain accessto the known monotonicity formulae.The first instance of Conjecture 1.3 to fall at non-conserved critical regularity was thecase s c = 1 / d = 3. This was achieved by Kenig and Merle, [14]. They usedthe concentration-compactness technique in the manner they pioneered in [13] together withthe Lin–Strauss Morawetz inequality.The present paper is motivated by a recent preprint of Kenig and Merle, [15], who considerspherically-symmetric solutions to a class of defocusing energy-supercritical nonlinear waveequations in three dimensions. They prove that if the solution is known to be uniformlybounded in the critical ˙ H sx -space throughout its lifetime, then the solution must be globaland it must scatter; this is precisely the NLW analogue of Conjecture 1.3.Earlier, we drew a parallel to the Navier–Stokes equation. The most natural analogue ofConjecture 1.3 in that setting is to show that boundedness of a critical norm implies globalregularity. Such results are known; see [9] and the references therein.In this paper we prove Conjecture 1.3 in several instances of defocusing energy-supercriticalnonlinear Schr¨odinger equations in dimensions d ≥ d ≥ F ( u ) = | u | u , and hence the critical regularity is s c = d − . Note that in dimension d ≥ d ≥
5, that is, p > d − . In this case, we alsoimpose some additional constraints on the power p . First, we ask that the nonlinearityobeys a certain smoothness condition; more precisely, we ask that s c < p , which isequivalent to 2 p − p ( d −
2) + 4 >
0. The role of this constraint is to allow us to take s c -many derivatives of the nonlinearity F ( u ); this is important in the development of thelocal theory. Moreover, in Section 6, we require that s c and p obey some further constraints.Together, these amount to ( < s c < for d = 5 , < s c < d +2 − √ ( d − − for d ≥ . (1.5) ROWAN KILLIP AND MONICA VISAN
One should not view (1.5) as a major constraint on the size of the critical regularity s c .Indeed, we claim that an interpolation of the techniques we present to treat the two problemsoutlined above can be used to treat any defocusing energy-supercritical NLS (with thesolution apriorily bounded in ˙ H s c x ) in dimensions d ≥
5, without any additional constrainton s c if the power p is an even integer and requiring merely the smoothness condition s c < p for arbitrary powers p . However, for the sake of readability, we chose not to workin this greater generality.Our main results are the following: Theorem 1.4 (Spacetime bounds – the cubic) . Let d ≥ and F ( u ) = | u | u . Let u : I × R d → C be a maximal-lifespan solution to (1.1) such that u ∈ L ∞ t ˙ H d − x ( I × R d ) . Then u is global and moreover, S R ( u ) ≤ C (cid:0) k u k L ∞ t ˙ H d − x (cid:1) . Theorem 1.5 (Spacetime bounds) . Let d ≥ and assume the critical regularity s c satisfies (1.5) . Let u : I × R d → C be a maximal-lifespan solution to (1.1) such that u ∈ L ∞ t ˙ H s c x ( I × R d ) . Then u is global and moreover, S R ( u ) ≤ C (cid:0) k u k L ∞ t ˙ H scx (cid:1) . As we already mentioned, the proofs of Theorems 1.4 and 1.5 follow closely the approachtaken in [19] to study the energy-critical problem. We outline the argument in subsection 1.1below. Our decision to work in dimensions d ≥ u for that equationmeans that it resembles the simpler cubic case treated here. We also believe that the argu-ments adapt to the corresponding energy-supercritical wave equations in dimensions d ≥ ≤ s c ≤
1, as well as more sophisticated stability results in the spirit of [23, 32].
Theorem 1.6 (Local well-posedness) . Let d and s c be as in Theorem 1.4 or 1.5. Then, given u ∈ ˙ H s c x ( R d ) and t ∈ R , there exists a unique maximal-lifespan solution u : I × R d → C to (1.1) with initial data u ( t ) = u . This solution also has the following properties: • (Local existence) I is an open neighbourhood of t . • (Blowup criterion) If sup I is finite, then u blows up forward in time (in the sense ofDefinition 1.2). If inf I is finite, then u blows up backward in time. • (Scattering) If sup I = + ∞ and u does not blow up forward in time, then u scattersforward in time, that is, there exists a unique u + ∈ ˙ H s c x ( R d ) such that lim t → + ∞ k u ( t ) − e it ∆ u + k ˙ H scx ( R d ) = 0 . (1.6) Conversely, given u + ∈ ˙ H s c x ( R d ) there is a unique solution to (1.1) in a neighbourhood ofinfinity so that (1.6) holds. • (Small data global existence) If (cid:13)(cid:13) |∇| s c u k is sufficiently small (depending on d, p ), then u is a global solution which does not blow up either forward or backward in time. Indeed, inthis case S R ( u ) . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) p ( d +2) / . NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 5 In Section 3 we establish this theorem as a corollary of our stability results Theorems 3.3and 3.4. These stability results are essential to the arguments we present, more specifically,to the proof of Theorem 1.12.1.1.
Outline of the proofs of Theorems 1.4 and 1.5.
We argue by contradiction. Weshow that if either Theorem 1.4 or Theorem 1.5 failed, this would imply the existence of avery special type of counterexample. Such counterexamples are then shown to have a wealthof properties not immediately apparent from their construction, so many properties, in fact,that they cannot exist.While we will make some further reductions later, the main property of the special coun-terexamples is almost periodicity modulo symmetries:
Definition 1.7 (Almost periodicity modulo symmetries) . Suppose s c >
0. A solution u to(1.1) with lifespan I is said to be almost periodic modulo symmetries if there exist functions N : I → R + , x : I → R d , and C : R + → R + such that for all t ∈ I and η > Z | x − x ( t ) |≥ C ( η ) /N ( t ) (cid:12)(cid:12) |∇| s c u ( t, x ) (cid:12)(cid:12) dx + Z | ξ |≥ C ( η ) N ( t ) | ξ | s c | ˆ u ( t, ξ ) | dξ ≤ η. We refer to the function N as the frequency scale function for the solution u , x the spatialcenter function , and to C as the compactness modulus function . Remark 1.8.
The parameter N ( t ) measures the frequency scale of the solution at time t , while 1 /N ( t ) measures the spatial scale. It is possible to multiply N ( t ) by any functionof t that is bounded both above and below, provided that we also modify the compactnessmodulus function C accordingly. Remark 1.9.
When s c = 0 the equation admits a new symmetry, namely, Galilei invari-ance. This introduces a frequency center function ξ ( t ) in the definition of almost periodicitymodulo symmetries; see [17, 34] for further discussion. Remark 1.10.
By the Ascoli–Arzela Theorem, a family of functions is precompact in˙ H s c x ( R d ) if and only if it is norm-bounded and there exists a compactness modulus function C so that Z | x |≥ C ( η ) (cid:12)(cid:12) |∇| s c f ( x ) (cid:12)(cid:12) dx + Z | ξ |≥ C ( η ) | ξ | s c | ˆ f ( ξ ) | dξ ≤ η for all functions f in the family. Thus, an equivalent formulation of Definition 1.7 is asfollows: u is almost periodic modulo symmetries if and only if { u ( t ) : t ∈ I } ⊆ { λ p f ( λ ( x + x )) : λ ∈ (0 , ∞ ) , x ∈ R d , and f ∈ K } for some compact subset K of ˙ H s c x ( R d ). Remark 1.11.
A further consequence of compactness modulo symmetries is the existenceof a function c : R + → R + so that Z | x − x ( t ) |≤ c ( η ) /N ( t ) (cid:12)(cid:12) |∇| s c u ( t, x ) (cid:12)(cid:12) dx + Z | ξ |≤ c ( η ) N ( t ) | ξ | s c | ˆ u ( t, ξ ) | dξ ≤ η for all t ∈ I and η > Theorem 1.12 (Reduction to almost periodic solutions) . Suppose that Theorem 1.4 (orTheorem 1.5) failed. Then there exists a maximal-lifespan solution u : I × R d → C to (1.1) such that u ∈ L ∞ t ˙ H s c x ( I × R d ) , u is almost periodic modulo symmetries, and u blows up bothforward and backward in time. Moreover, u has minimal L ∞ t ˙ H s c x -norm among all blowupsolutions, that is, sup t ∈ I (cid:13)(cid:13) |∇| s c u ( t ) (cid:13)(cid:13) ≤ sup t ∈ J (cid:13)(cid:13) |∇| s c v ( t ) (cid:13)(cid:13) for all maximal-lifespan solutions v : J × R d → C that blow up in at least one time direction. The reduction to almost periodic solutions was first realized by Keraani [17] in the contextof the mass-critical NLS. This was adapted to the energy-critical case by Kenig and Merle[13].Theorem 1.12 provides no information on the modulation parameters x ( t ) and N ( t ). Inorder to treat the mass-critical NLS in two dimensions, a further reduction was introducedin [18] where the behaviour of N ( t ) is heavily constrained. This argument wa adapted tothe energy-critical case in [19]. This latter argument is directly applicable to the setting ofthis paper and yields Theorem 1.13 (Three special scenarios for blowup) . Suppose that Theorem 1.4 (or Theo-rem 1.5) failed. Then there exists a maximal-lifespan solution u : I × R d → C , which obeys u ∈ L ∞ t ˙ H s c x ( I × R d ) , is almost periodic modulo symmetries, and S I ( u ) = ∞ . Moreover, wecan also ensure that the lifespan I and the frequency scale function N : I → R + match oneof the following three scenarios: I. (Finite-time blowup) We have that either sup I < ∞ or | inf I | < ∞ . II. (Soliton-like solution) We have I = R and N ( t ) = 1 for all t ∈ R . III. (Low-to-high frequency cascade) We have I = R , inf t ∈ R N ( t ) ≥ , and lim sup t → + ∞ N ( t ) = ∞ . Therefore, in order to prove Theorems 1.4 and 1.5 it suffices to preclude the existence ofsolutions that satisfy the criteria in Theorem 1.13. Following [19], the key step in all threescenarios above is to prove negative regularity, that is, the solution u lies in L x or better.In scenarios II and III, the proof that u ∈ L x requires d ≥
5; note that this expresses betterdecay of the solution at spatial infinity. Similar in spirit to [18, 19, 21], negative regularityis deduced (via almost periodicity) from the minimality of the solution considered; recallthat u has minimal L ∞ t ˙ H s c x norm among all blowup solutions.A further manifestation of this minimality is the absence of a scattered wave at theendpoints of the lifespan I ; more formally, we have the following Duhamel formulae, whichplay an important role in proving negative regularity. For a proof, see [34, Section 6] or [20,Section 5]. Lemma 1.14 (No waste Duhamel formulae) . Let u be an almost periodic solution to (1.1) on its maximal-lifespan I . Then, for all t ∈ I , u ( t ) = lim T ր sup I i Z Tt e i ( t − t ′ )∆ F ( u ( t ′ )) dt ′ = − lim T ց inf I i Z tT e i ( t − t ′ )∆ F ( u ( t ′ )) dt ′ , (1.7) as weak limits in ˙ H s c x . We preclude the finite-time blowup scenario in Section 5. In order to achieve this, weuse an argument based on the Strichartz inequality and we take advantage of the finitenessof the blowup time. We prove that in this case, the solution must have zero mass/energy.This contradicts the fact that the solution blows up.To preclude the remaining two scenarios, we follow closely the strategy in [19]. Asdescribed above, the main point is to prove additional decay/negative regularity; this isachieved in two steps. First, we prove that the solution belongs to L ∞ t L qx for certain valuesof q less than dp/
2, which is the exponent obtained by applying Sobolev embedding to˙ H s c x . Just as in [19], the proof of this first step involves a bootstrap argument built off the NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 7 Duhamel formulae (1.7). In order to disentangle frequency interactions, we make use ofan ‘acausal’ Gronwall inequality, Lemma 2.13. In the second step, we upgrade this breachof scaling to negative regularity in L x -based spaces. To do this, we take advantage of theglobal existence together with a double Duhamel trick in the spirit of [19, 29]. In order tomake the associated time integrals converge, we need both d ≥ Acknowledgements.
The first author was supported by NSF grant DMS-0701085.2.
Notations and useful lemmas
Some notation.
We write X . Y or Y & X whenever X ≤ CY for some constant C >
0. We use O ( Y ) to denote any quantity X such that | X | . Y . We use the notation X ∼ Y to mean X . Y . X . The fact that these constants depend upon the dimension d or the power p will be suppressed. If C depends upon some additional parameters, we willindicate this with subscripts; for example, X . u Y denotes the assertion that X ≤ C u Y forsome C u depending on u ; similarly for X ∼ u Y , X = O u ( Y ), etc. We denote by X ± anyquantity of the form X ± ε for any ε > I × R d , we use L qt L rx ( I × R d ) to denote the Banach space offunctions u : I × R d → C whose norm is k u k L qt L rx ( I × R d ) := (cid:16)Z I k u ( t ) k qL rx dt (cid:17) q < ∞ , with the usual modifications when q or r is equal to infinity. When q = r we abbreviate L qt L qx as L qt,x .We define the Fourier transform on R d byˆ f ( ξ ) := (2 π ) − d/ Z R d e − ixξ f ( x ) dx. For s ∈ R , we define the fractional differentiation/integral operator \ |∇| s f ( ξ ) := | ξ | s ˆ f ( ξ ) , which in turn defines the homogeneous Sobolev norm k f k ˙ H sx ( R d ) := (cid:13)(cid:13) |∇| s f (cid:13)(cid:13) L x ( R d ) . Finally, we use the notation ∇ F ( u ( x )) := ∇ u ( x ) · F ′ ( u ( x )) := ∇ u ( x ) F z ( u ( x )) + ∇ u ( x ) F ¯ z ( u ( x )) , where F z , F ¯ z denote the usual complex derivatives F z := 12 (cid:16) ∂F∂x − i ∂F∂y (cid:17) , F ¯ z := 12 (cid:16) ∂F∂x + i ∂F∂y (cid:17) . Strichartz estimates.
Let e it ∆ be the free Schr¨odinger evolution. From the explicitformula e it ∆ f ( x ) = 1(4 πit ) d/ Z R d e i | x − y | / t f ( y ) dy, one easily obtains the standard dispersive inequality k e it ∆ f k L ∞ x ( R d ) . | t | − d k f k L x ( R d ) (2.1) ROWAN KILLIP AND MONICA VISAN for all t = 0. In particular, as the free propagator conserves the L x -norm, k e it ∆ f k L px ( R d ) . | t | d ( p − ) k f k L p ′ x ( R d ) (2.2)for all t = 0 and 2 ≤ p ≤ ∞ , where p + p ′ = 1. Definition 2.1 (Admissible pairs) . For d ≥
3, we say that a pair of exponents ( q, r ) is
Schr¨odinger-admissible if 2 q + dr = d ≤ q, r ≤ ∞ . (2.3)For a fixed spacetime slab I × R d , we define the Strichartz norm k u k S ( I ) := sup ( q,r ) admissible k u k L qt L rx ( I × R d ) . We write S ( I ) for the closure of all test functions under this norm and denote by N ( I )the dual of S ( I ).A simple application of Sobolev embedding yields (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L ∞ t L x ( I × R d ) + (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L d +2) dt,x ( I × R d ) + k u k L ∞ t L dp x ( I × R d ) + k u k L p ( d +2)2 t,x ( I × R d ) . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) for all d ≥ Lemma 2.2 (Strichartz) . Let s ≥ , let I be a compact time interval, and let u : I × R d → C be a solution to the forced Schr¨odinger equation iu t + ∆ u = F. Then, (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) S ( I ) . (cid:13)(cid:13) |∇| s u ( t ) (cid:13)(cid:13) L x + (cid:13)(cid:13) |∇| s F (cid:13)(cid:13) N ( I ) for any t ∈ I .Proof. See, for example, [10, 27]. For the endpoint ( q, r ) = (cid:0) , dd − (cid:1) in dimensions d ≥ (cid:3) Basic harmonic analysis.
Let ϕ ( ξ ) be a radial bump function supported in the ball { ξ ∈ R d : | ξ | ≤ } and equal to 1 on the ball { ξ ∈ R d : | ξ | ≤ } . For each number N > \ P ≤ N f ( ξ ) := d f ≤ N ( ξ ) := ϕ ( ξ/N ) ˆ f ( ξ ) \ P >N f ( ξ ) := d f >N ( ξ ) := (1 − ϕ ( ξ/N )) ˆ f ( ξ ) d P N f ( ξ ) := c f N ( ξ ) := ( ϕ ( ξ/N ) − ϕ (2 ξ/N )) ˆ f ( ξ )and similarly P
Corollary 2.7.
Let F ( u ) = | u | p u and let s ≥ if p is an even integer or ≤ s < p otherwise. Then, on any spacetime slab I × R d we have (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) N ( I ) . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) S ( I ) k u k pL p ( d +2)2 t,x and (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L dd +4 x . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) L ∞ t L x k u k pL ∞ t L dp/ x . Revisiting the proof of Lemma 2.6, we obtain the following lemma. In actually, the resultcan de deduced directly from (2.4) and Lemma 2.4; however, the result (2.6) appearing inthe proof will be needed in Section 4.
Lemma 2.8.
Let G be a H¨older continuous function of order < p ≤ and let < s <σp < p . For < q, q , q , r , r , r < ∞ such that q = q + q = r + r + r we have (cid:13)(cid:13) |∇| s (cid:2) w · (cid:0) G ( u + v ) − G ( u ) (cid:1)(cid:3)(cid:13)(cid:13) q . (cid:13)(cid:13) |∇| s w (cid:13)(cid:13) q k v k ppq + k w k r k v k p − sσ ( p − sσ ) r (cid:0)(cid:13)(cid:13) |∇| σ v (cid:13)(cid:13) sσ r + (cid:13)(cid:13) |∇| σ u (cid:13)(cid:13) sσ r (cid:1) sσ , provided (1 − p ) r , ( p − sσ ) r > .Proof. In [26], Strichartz proved that for all Schwartz functions f , 1 < q < ∞ , and 0 < s < (cid:13)(cid:13) |∇| s f (cid:13)(cid:13) L qx ∼ (cid:13)(cid:13) D s ( f ) (cid:13)(cid:13) L qx , where D s ( f )( x ) := (cid:18)Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z | y | < (cid:12)(cid:12) f ( x + ry ) − f ( x ) (cid:12)(cid:12) dy (cid:12)(cid:12)(cid:12)(cid:12) drr s (cid:19) / . (2.5)In view of this, the claim will follow from the pointwise inequality D s (cid:0) w · (cid:2) G ( u + v ) − G ( u ) (cid:3)(cid:1) (2.6) . D s ( w ) | v | p + (cid:2) M ( | w | − p ) (cid:3) − p (cid:2) M ( | v | ) (cid:3) p − sσ (cid:2) D σ ( u + v ) + D σ ( u ) (cid:3) sσ , where M denotes the Hardy–Littlewood maximal function.As G is H¨older continuous of order p , (cid:12)(cid:12)(cid:0) w · (cid:2) G ( u + v ) − G ( u ) (cid:3)(cid:1) ( x + ry ) − (cid:0) w · (cid:2) G ( u + v ) − G ( u ) (cid:3)(cid:1) ( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) w ( x + ry ) − w ( x ) (cid:12)(cid:12)(cid:12)(cid:12) G ( u + v )( x ) − G ( u )( x ) (cid:12)(cid:12) + (cid:12)(cid:12) w ( x + ry ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) G ( u + v ) − G ( u ) (cid:3) ( x + ry ) − (cid:2) G ( u + v ) − G ( u ) (cid:3) ( x ) (cid:12)(cid:12) . (cid:12)(cid:12) w ( x + ry ) − w ( x ) (cid:12)(cid:12) | v ( x ) | p + H ( x ) (2.7)with H := (cid:12)(cid:12) w ( x + ry ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) G ( u + v ) − G ( u ) (cid:3) ( x + ry ) − (cid:2) G ( u + v ) − G ( u ) (cid:3) ( x ) (cid:12)(cid:12) . (2.8)Note that the first term on the right-hand side of (2.7) gives rise to the term D s ( w ) | v | p in (2.6). Hence it remains to estimate the contribution of H . In order to achieve this, weestimate H in two different ways: H . (cid:12)(cid:12) w ( x + ry ) (cid:12)(cid:12)h | v ( x + ry ) | p + | v ( x ) | p i (2.9)and also H . (cid:12)(cid:12) w ( x + ry ) (cid:12)(cid:12)h(cid:12)(cid:12) ( u + v )( x + ry ) − ( u + v )( x ) (cid:12)(cid:12) p + (cid:12)(cid:12) u ( x + ry ) − u ( x ) (cid:12)(cid:12) p i . (2.10)Using H¨older’s inequality and (2.9), we see that Z ∞ A ( x ) (cid:12)(cid:12)(cid:12)(cid:12)Z | y | < H dy (cid:12)(cid:12)(cid:12)(cid:12) drr s . (cid:2) M ( | w | − p )( x ) (cid:3) − p ) (cid:2) M ( | v | )( x ) (cid:3) p Z ∞ A ( x ) drr s . (cid:2) A ( x ) (cid:3) − s (cid:2) M ( | w | − p )( x ) (cid:3) − p ) (cid:2) M ( | v | )( x ) (cid:3) p . The precise value of A ( x ) will be determined below.We now turn our attention to small values of r . Using (2.10) together with H¨older’sinequality, we see that Z | y | < H dy . (cid:2) M (cid:0) | w | − p (cid:1) ( x ) (cid:3) − p (cid:20)Z | y | < (cid:12)(cid:12) ( u + v )( x + ry ) − ( u + v )( x ) (cid:12)(cid:12) + (cid:12)(cid:12) u ( x + ry ) − u ( x ) (cid:12)(cid:12) dy (cid:21) p and so, applying H¨older’s inequality again, we find Z A ( x )0 (cid:12)(cid:12)(cid:12)(cid:12)Z | y | < H dy (cid:12)(cid:12)(cid:12)(cid:12) drr s . (cid:2) M (cid:0) | w | − p (cid:1) ( x ) (cid:3) − p ) (cid:2) A ( x ) (cid:3) σp − s ) (cid:2) D σ ( u + v )( x ) + D σ ( u )( x ) (cid:3) p . Putting things together and optimizing the choice of A ( x ), we derive (2.6). This finishesthe proof of the lemma. (cid:3) The next result is formally similar to Lemma 2.6. The proof is simple; see the appendixin [20]. It is used in the proof of Lemma 6.2.
NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 11 Lemma 2.9 (Nonlinear Bernstein, [20]) . Let G : C → C be H¨older continuous of order < p ≤ . Then k P N G ( u ) k L q/px ( R d ) . N − p k∇ u k pL qx ( R d ) for any ≤ q < ∞ . Concentration compactness.
In this subsection we record the linear profile decom-position statement which will lead to the reduction in Theorem 1.12. We first recall thesymmetries of the equation (1.1) which fix the initial surface t = 0. Definition 2.10 (Symmetry group) . For any phase θ ∈ R / π Z , position x ∈ R d , andscaling parameter λ >
0, we define a unitary transformation g θ,x ,λ : ˙ H s c x ( R d ) → ˙ H s c x ( R d )by [ g θ,x ,λ f ]( x ) := λ − p e iθ f (cid:0) λ − ( x − x ) (cid:1) . Recall that s c := d − p . Let G denote the collection of such transformations. For a function u : I × R d → C , we define T g θ,x ,λ u : λ I × R d → C where λ I := { λ t : t ∈ I } by theformula [ T g θ,x ,λ u ]( t, x ) := λ − p e iθ u (cid:0) λ − t, λ − ( x − x ) (cid:1) . Note that if u is a solution to (1.1), then T g u is a solution to (1.1) with initial data gu . Remark 2.11.
It is easy to verify that G is a group and that the map g T g is ahomomorphism. The map u T g u maps solutions to (1.1) to solutions with the samescattering size as u , that is, S ( T g u ) = S ( u ). Furthermore, u is a maximal-lifespan solutionif and only if T g u is a maximal-lifespan solution.We are now ready to state the linear profile decomposition; in the generality needed here,this was proved in [25]. For s c = 0 the linear profile decomposition was proved in [1, 3, 22],while for s c = 1 it was established in [16]. Lemma 2.12 (Linear profile decomposition, [25]) . Fix s c > and let { u n } n ≥ be a sequenceof functions bounded in ˙ H s c x ( R d ) . Then, after passing to a subsequence if necessary, thereexist functions { φ j } j ≥ ⊂ ˙ H s c x ( R d ) , group elements g jn ∈ G , and times t jn ∈ R such that forall J ≥ we have the decomposition u n = J X j =1 g jn e it jn ∆ φ j + w Jn with the following properties: • w Jn ∈ ˙ H s c x ( R d ) and obey lim J →∞ lim sup n →∞ (cid:13)(cid:13) e it ∆ w Jn (cid:13)(cid:13) L p ( d +2)2 t,x ( R × R d ) = 0 . • For any j = j ′ , λ jn λ j ′ n + λ j ′ n λ jn + | x jn − x j ′ n | λ jn λ j ′ n + (cid:12)(cid:12) t jn ( λ jn ) − t j ′ n ( λ j ′ n ) (cid:12)(cid:12) λ jn λ j ′ n → ∞ as n → ∞ . • For any J ≥ we have the decoupling properties: lim n →∞ h(cid:13)(cid:13) |∇| s c u n (cid:13)(cid:13) − J X j =1 (cid:13)(cid:13) |∇| s c φ j (cid:13)(cid:13) − (cid:13)(cid:13) |∇| s c w Jn (cid:13)(cid:13) i = 0 and for any ≤ j ≤ J , e − it jn ∆ [( g jn ) − w Jn ] → weakly in ˙ H s c x as n → ∞ . A Gronwall inequality.
Our last technical tool is a form of Gronwall’s inequalitythat involves both the past and the future, ‘acausal’ in the terminology of [30]. We importit from [19], where it was used for precisely the same purpose as it will be here.
Lemma 2.13 (Acausal Gronwall inequality, [19]) . Given γ > , < η < (1 − − γ ) , and { b k } ∈ ℓ ∞ ( Z + ) , let x k ∈ ℓ ∞ ( Z + ) be a non-negative sequence obeying x k ≤ b k + η ∞ X l =0 − γ | k − l | x l for all k ≥ .Then x k . k X l =0 r | k − l | b l for all k ≥ for some r = r ( η ) ∈ (2 − γ , . Moreover, r ↓ − γ as η ↓ . Local well-posedness
In this section we develop the local well-posedness theory for (1.1). The arguments we useare inspired by previous work on nonlinear Schr¨odinger equations at critical regularity. For s c ∈ [0 ,
1] the standard local well-posedness theory (see Theorem 3.1 below) was establishedby Cazenave and Weissler, [4]; see also [5, 30]. For stability results (see Theorems 3.3 and3.4 below) in the mass- and energy-critical settings (i.e. s c = 0 , Theorem 3.1 (Standard local well-posedness) . Let d ≥ , s c ≥ , and let u ∈ H s c x ( R d ) .Assume in addition that s c < p if p is not an even integer. Then there exists η = η ( d ) > such that if < η ≤ η and I is a compact interval containing zero such that (cid:13)(cid:13) |∇| s c e it ∆ u (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ η, (3.1) then there exists a unique solution u to (1.1) on I × R d . Moreover, we have the bounds (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ η (3.2) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I × R d ) . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L x + η p (3.3) k u k S ( I × R d ) . k u k L x . (3.4) Remark 3.2.
By the Strichartz inequality, we know that (cid:13)(cid:13) |∇| s c e it ∆ u (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( R × R d ) . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L x . Thus, (3.1) holds with I = R for initial data with sufficiently small norm. Alternatively,by the monotone convergence theorem, (3.1) holds provided I is chosen sufficiently small.Note that by scaling, the length of the interval I depends on the fine properties of u , notonly on its norm. Proof.
We will essentially repeat the standard argument from [4]; the fractional chain ruleLemma 2.5 leads to some simplifications.
NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 13 The theorem follows from a contraction mapping argument. More precisely, using theStrichartz estimates from Lemma 2.2, we will show that the map u Φ( u ) defined byΦ( u )( t ) := e it ∆ u − i Z t e i ( t − s )∆ F ( u ( s )) ds, is a contraction on the set B ∩ B where B := n u ∈ L ∞ t H s c x ( I × R d ) : k u k L ∞ t H scx ( I × R d ) ≤ k u k H scx + C ( d, p )(2 η ) p o B := n u ∈ L p +2 t W s c , d ( p +2)2( d − dp x ( I × R d ) : (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ η and (cid:13)(cid:13) u (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ C ( d, p ) k u k L x o under the metric given by d ( u, v ) := k u − v k L p +2 t L d ( p +2)2( d − dpx ( I × R d ) . Here C ( d, p ) denotes a constant that changes from line to line. Note that the norm appearingin the metric scales like L x . Note also that both B and B are closed (and hence complete)in this metric.Using the Strichartz inequality followed by Corollary 2.7 and Sobolev embedding, we findthat for u ∈ B ∩ B , k Φ( u ) k L ∞ t H scx ( I × R d ) ≤ k u k H scx + C ( d, p ) (cid:13)(cid:13) h∇i s c F ( u ) (cid:13)(cid:13) L p +2 p +1 t L d ( p +2)2( d +2)+ dpx ( I × R d ) ≤ k u k H scx + C ( d, p ) (cid:13)(cid:13) h∇i s c u (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) k u k pL p +2 t L dp ( p +2)4 x ( I × R d ) ≤ k u k H scx + C ( d, p ) (cid:0) η + 2 C ( d, p ) k u k L x (cid:1)(cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) pL p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ k u k H scx + C ( d, p ) (cid:0) η + 2 C ( d, p ) k u k L x (cid:1) (2 η ) p and similarly, (cid:13)(cid:13) Φ( u ) (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ C ( d, p ) k u k L x + C ( d, p ) (cid:13)(cid:13) F ( u ) (cid:13)(cid:13) L p +2 p +1 t L d ( p +2)2( d +2)+ dpx ( I × R d ) ≤ C ( d, p ) k u k L x + C ( d, p ) k u k L x (2 η ) p . Arguing as above and invoking (3.1), we obtain (cid:13)(cid:13) |∇| s c Φ( u ) (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ η + C ( d, p ) (cid:13)(cid:13) |∇| s c F ( u ) (cid:13)(cid:13) L p +2 p +1 t L d ( p +2)2( d +2)+ dpx ( I × R d ) ≤ η + C ( d, p )(2 η ) p . Thus, choosing η = η ( d ) sufficiently small, we see that for 0 < η ≤ η , the functionalΦ maps the set B ∩ B back to itself. To see that Φ is a contraction, we repeat thecomputations above to obtain (cid:13)(cid:13) Φ( u ) − Φ( v ) (cid:13)(cid:13) L p +2 t L d ( p +2)2( d − dpx ( I × R d ) ≤ C ( d, p ) (cid:13)(cid:13) F ( u ) − F ( v ) (cid:13)(cid:13) L p +2 p +1 t L d ( p +2)2( d +2)+ dpx ( I × R d ) ≤ C ( d, p )(2 η ) p k u − v k L p +2 t L d ( p +2)2( d − dpx ( I × R d ) . Thus, choosing η = η ( d ) even smaller (if necessary), we can guarantee that Φ is a contrac-tion on the set B ∩ B . By the contraction mapping theorem, it follows that Φ has a fixedpoint in B ∩ B . Moreover, noting that Φ maps into C t H s c x (not just L ∞ t H s c x ), we derive(after one more application of the Strichartz inequality) that the fixed point of Φ is indeeda solution to (1.1). We now turn our attention to the uniqueness statement. Since uniqueness is a local prop-erty, it suffices to study a neighbourhood of t = 0. By Definition 1.1 (and the Strichartzinequality), any solution to (1.1) belongs to B ∩ B on some such neighbourhood. Unique-ness thus follows from uniqueness in the contraction mapping theorem.The claims (3.3) and (3.4) follow from another application of the Strichartz inequality,as above. (cid:3) Next, we will establish a stability theory for (1.1) in the settings of Theorems 1.4 and 1.5.We start with the cubic NLS.
Theorem 3.3 (Stability – the cubic) . Let d ≥ and let I a compact time interval containingzero and ˜ u be an approximate solution to (1.1) on I × R d in the sense that i ˜ u t = − ∆˜ u + | ˜ u | ˜ u + e for some function e . Assume that k ˜ u k L ∞ t ˙ H d − x ( I × R d ) ≤ E (3.5) S I (˜ u ) ≤ L (3.6) for some positive constants E and L . Let u ∈ ˙ H d − x and assume the smallness conditions k u − ˜ u k ˙ H d − x ≤ ε (3.7) (cid:13)(cid:13) |∇| d − e (cid:13)(cid:13) N ( I ) ≤ ε (3.8) for some < ε < ε = ε ( E, L ) . Then, there exists a unique strong solution u : I × R d C to (1.1) with initial data u at time t = 0 satisfying S I ( u − ˜ u ) ≤ C ( E, L ) ε d +2 (3.9) (cid:13)(cid:13) |∇| d − ( u − ˜ u ) (cid:13)(cid:13) S ( I ) ≤ C ( E, L ) ε (3.10) (cid:13)(cid:13) |∇| d − u (cid:13)(cid:13) S ( I ) ≤ C ( E, L ) . (3.11) Proof.
We will prove the theorem under the additional assumption that u ∈ L x , so thatwe can rely on Theorem 3.1 to guarantee that u exists. This additional assumption can beremoved a posteriori by the usual limiting argument: approximate u in ˙ H s c x by { u n (0) } n ⊆ H s c x and apply the theorem with ˜ u = u m , u = u n , and e = 0 to deduce that the sequence ofsolutions { u n } n with initial data { u n (0) } n is Cauchy in critical norms and thus convergentto a solution u with initial data u which obeys |∇| s c u ∈ S ( I ). Thus, it suffices to prove(3.9) through (3.11) as a priori estimates, that is we assume that the solution u exists andobeys |∇| s c u ∈ S ( I ).We first prove (3.9) through (3.11) under the stronger hypothesis that (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) L d +2 t L d ( d +2) d d − x ( I × R d ) ≤ δ (3.12)for some small 0 < δ = δ ( d, p ).Let w := u − ˜ u . Then w satisfies the following initial value problem ( iw t = − ∆ w + F (˜ u + w ) − F (˜ u ) − ew (0) = u − ˜ u . For t ∈ I we define A ( t ) := (cid:13)(cid:13) |∇| d − (cid:2) F (˜ u + w ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ([0 ,t ]) . NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 15 By the (fractional) chain rule and (3.12), A ( t ) . (cid:13)(cid:13) |∇| d − w (cid:13)(cid:13) S ( I ) + δ (cid:13)(cid:13) |∇| d − w (cid:13)(cid:13) S ( I ) + δ (cid:13)(cid:13) |∇| d − w (cid:13)(cid:13) S ( I ) . (3.13)On the other hand, by Strichartz, (3.7), and (3.8), we get (cid:13)(cid:13) |∇| d − w (cid:13)(cid:13) S ( I ) . k u − ˜ u k ˙ H d − x + A ( t ) + (cid:13)(cid:13) |∇| d − e (cid:13)(cid:13) N ( I ) . A ( t ) + ε. (3.14)Combining (3.13) and (3.14), we obtain A ( t ) . ( A ( t ) + ε ) + δ ( A ( t ) + ε ) + δ ( A ( t ) + ε ) + ε. A standard continuity argument then shows that if δ is taken sufficiently small, A ( t ) . ε for any t ∈ I, which immediately implies (3.9) through (3.11) via an application of the Strichartz inequalityand the triangle inequality.We now prove (3.9) through (3.11) under the hypothesis (3.6), as opposed to (3.12). Wefirst show that (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) S ( I ) ≤ C ( E, L ) . (3.15)Indeed, by (3.6) we may divide I into J = J ( L, η ) subintervals I j = [ t j , t j +1 ] such that oneach spacetime slab I j × R d k ˜ u k L d +2 t,x ( I j × R d ) ≤ η for a small constant η > (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) S ( I j ) . k ˜ u ( t j ) k ˙ H d − x + (cid:13)(cid:13) |∇| d − e (cid:13)(cid:13) N ( I j ) + (cid:13)(cid:13) |∇| d − F (˜ u ) (cid:13)(cid:13) N ( I j ) . E + ε + k ˜ u k L d +2 t,x ( I j × R d ) (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) S ( I j ) . E + ε + η (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) S ( I j ) . Thus, choosing η > d and ε sufficiently small depending on E , weobtain (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) S ( I j ) . E. Summing this over all subintervals I j , we derive (3.15). Thus, we may divide I into J = J ( E, L ) subintervals I j = [ t j , t j +1 ] such that on each spacetime slab I j × R d (cid:13)(cid:13) |∇| d − ˜ u (cid:13)(cid:13) L d +2 t L d ( d +2) d d − x ( I j × R d ) ≤ δ for some small δ = δ ( d, p ) > ε sufficiently small (depending on J ), we can iterate the argument above toobtain for each 0 ≤ j < J and all 0 < ε < ε , S I j ( u − ˜ u ) ≤ C ( j ) ε d +2 (cid:13)(cid:13) |∇| d − ( u − ˜ u ) (cid:13)(cid:13) S ( I j ) ≤ C ( j ) ε (cid:13)(cid:13) |∇| d − u (cid:13)(cid:13) S ( I j ) ≤ C ( j ) E (cid:13)(cid:13) |∇| d − (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ( I j ) ≤ C ( j ) ε, (3.16)provided we can show k u ( t j ) − ˜ u ( t j ) k ˙ H d − x ≤ C ( j − ε (3.17) for each 1 ≤ j < J . By the Strichartz inequality and the inductive hypothesis, k u ( t j ) − ˜ u ( t j ) k ˙ H d − x . k u − ˜ u k ˙ H d − x + (cid:13)(cid:13) |∇| d − e (cid:13)(cid:13) N ([0 ,t j ]) + (cid:13)(cid:13) |∇| d − (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ([0 ,t j ]) . ε + j − X k =0 C ( k ) ε, which proves (3.17).Summing the bounds in (3.16) over all subintervals I j , we derive (3.9) through (3.11).This completes the proof of the theorem. (cid:3) We now address the stability question in the setting of Theorem 1.5. We will prove thefollowing result.
Theorem 3.4 (Stability) . Let d ≥ and assume the critical regularity s c satisfies (1.5) .Let I a compact time interval containing zero and let ˜ u be an approximate solution to (1.1) on I × R d in the sense that i ˜ u t = − ∆˜ u + F (˜ u ) + e for some function e . Assume that k ˜ u k L ∞ t ˙ H scx ( I × R d ) ≤ E (3.18) S I (˜ u ) ≤ L (3.19) for some positive constants E and L . Let u ∈ ˙ H s c x and assume the smallness conditions k u − ˜ u k ˙ H scx ≤ ε (3.20) (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I ) ≤ ε (3.21) for some < ε < ε = ε ( E, L ) . Then, there exists a unique strong solution u : I × R d C to (1.1) with initial data u at time t = 0 satisfying S I ( u − ˜ u ) ≤ C ( E, L ) ε c (3.22) (cid:13)(cid:13) |∇| s c ( u − ˜ u ) (cid:13)(cid:13) S ( I ) ≤ C ( E, L ) ε c (3.23) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) ≤ C ( E, L ) , (3.24) where c , c are positive constants that depend on d, p, E, and L . Remarks.
1. Theorems 3.3 and 3.4 imply the existence and uniqueness of maximal-lifespansolutions in Theorem 3.1. They also prove that the solutions depend uniformly continuouslyon the initial data (on bounded sets) in norms which are critical with respect to scaling.As a consequence, one can remove from Theorem 3.1 the assumption that the initial databelongs to L x , since every ˙ H s c x function is well approximated by H s c x functions.2. Using Theorem 3.1 (without the additional assumption that u ∈ L x , due to the firstpoint above), as well as its proof, one easily derives Theorem 1.6. We omit the standarddetails.We now turn to the proof of Theorem 3.4; the argument we present is inspired by theone used in the energy-critical setting [20, 32]; see also [23] for a similar technique in thecontext of the Klein–Gordon equation. The idea is to work in spaces which are critical withrespect to scaling but have a small fractional number of derivatives. NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 17 For the remainder of this subsection, for any time interval I we will use the abbreviations k u k X ( I ) := k u k L q t L r dd − r scx ( I × R d ) k u k X ( I ) := (cid:13)(cid:13) |∇| p/ u (cid:13)(cid:13) L q t L r d d − r sc − p ) x ( I × R d ) k F k Y ( I ) := (cid:13)(cid:13) |∇| p/ F (cid:13)(cid:13) L q pt L r d p )( d − r sc )+ r px ( I × R d ) , (3.25)where ( q , r ) = (cid:0) p (2+ p ) p − p ( d − , d (2+ p ) d − p + s c (2+ p ) (cid:1) is a Schr¨odinger admissible pair. Note thatbecause of (1.5) we have 2 < r < ds c and p ( d +2)2 < q < ∞ .First, we connect the spaces in which the solution to (1.1) is measured to the spaces inwhich the nonlinearity is measured. As usual, this is done via a Strichartz inequality; wereproduce the standard proof. Lemma 3.5 (Strichartz estimate) . Let I be a compact time interval containing t . Then (cid:13)(cid:13)(cid:13)Z tt e i ( t − s )∆ F ( s ) ds (cid:13)(cid:13)(cid:13) X ( I ) . k F k Y ( I ) . Proof. As r d d − r (2 s c − p ) and r d p )( d − r s c )+ r p are dual exponents, the dispersive estimate(2.2) implies (cid:13)(cid:13) e i ( t − s )∆ F ( s ) (cid:13)(cid:13) L r d d − r sc − p ) x . | t − s | − p ( d − r sc )2 r k F ( s ) k L r d p )( d − r sc )+ r px . Using the Hardy-Littlewood-Sobolev inequality and the fact that ( q , r ) is a Schr¨odingeradmissible pair, we obtain (cid:13)(cid:13)(cid:13)Z tt e i ( t − s )∆ F ( s ) ds (cid:13)(cid:13)(cid:13) L q t L r d d − r sc − p ) x ( I × R d ) . k F k L q pt L r d p )( d − r sc )+ r px ( I × R d ) . As the differentiation operator |∇| p/ commutes with the free evolution, we recover theclaim. (cid:3) Next we establish some connections between the spaces defined in (3.25) and the usualStrichartz spaces.
Lemma 3.6 (Interpolations) . For any compact time interval I , k u k X ( I ) . k u k X ( I ) . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) (3.26) k u k X ( I ) . k u k θ L p ( d +2)2 t,x ( I × R d ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) − θ S ( I ) (3.27) k u k L p ( d +2)2 t,x ( I × R d ) . k u k θ X ( I ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) − θ S ( I ) , (3.28) where < θ , θ < depend on d, p .Proof. A simple application of Sobolev embedding yields (3.26).Using interpolation, we obtain k u k X ( I ) . k u k − p sc X ( I ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) p sc L q t L r x ( I × R d ) . k u k − p sc X ( I ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) p sc S ( I ) . On the other hand, as p ( d +2)2 < q < ∞ , interpolation followed by Sobolev embedding yields k u k X ( I ) . k u k p ( d +2)2 q L p ( d +2)2 t,x ( I × R d ) k u k − p ( d +2)2 q L ∞ t L pd x ( I × R d ) . k u k p ( d +2)2 q L p ( d +2)2 t,x ( I × R d ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) − p ( d +2)2 q S ( I ) . Putting everything together, we derive (3.27).
We now turn to (3.28); using interpolation once again, we obtain k u k L p ( d +2)2 t,x ( I × R d ) . k u k q pd − p [ q d − d +2)] X ( I ) k u k − q pd − p [ q d − d +2)] L d +2) dt L pd ( d +2)2( d +2) − pdx ( I × R d ) and the claim follows from (3.26) and Sobolev embedding. (cid:3) Finally, we derive estimates that will help us control the nonlinearity. The main tools weuse in deriving these estimates are the fractional chain rules, Lemmas 2.5 and 2.6.
Lemma 3.7 (Nonlinear estimates) . Let d ≥ and assume the critical regularity s c satisfies (1.5) . Let I a compact time interval. Then, k F ( u ) k Y ( I ) . k u k p +1 X ( I ) , (3.29) k F z ( u + v ) w k Y ( I ) + k F ¯ z ( u + v ) ¯ w k Y ( I ) . (cid:16) k u k p ( sc − sc X ( I ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) psc S ( I ) + k v k p ( sc − sc X ( I ) (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) psc S ( I ) (cid:17) k w k X ( I ) , (3.30) and (cid:13)(cid:13) |∇| s c [ F ( u + v ) − F ( u )] (cid:13)(cid:13) N ( I ) (3.31) . (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) S ( I ) h k v k pX ( I ) + k u k p − sc X ( I ) k v k − sc X ( I ) + (cid:0) k u k p − sc X ( I ) + k v k p − sc X ( I ) (cid:1)(cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) − sc S ( I ) i + (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) (cid:16) k v k pX ( I ) + k u k βX ( I ) k v k p − βX ( I ) (cid:17) + (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) sc S ( I ) (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) − sc S ( I ) k u k − sc X ( I ) k v k p − sc X ( I ) for some < β < p .Proof. Throughout the proof, all spacetime norms are on I × R d .Applying Lemma 2.5 followed (3.26), we find k F ( u ) k Y ( I ) . k u k pX ( I ) k u k X ( I ) . k u k pX ( I ) . This establishes (3.29).We now turn to (3.30); we only treat the first term on the left-hand side, as the secondterm can be handled similarly. By Lemma 2.4 followed by (3.26), k F z ( u + v ) w k Y ( I ) . k F z ( u + v ) k L q pt L r dp ( d − r sc ) x k w k X ( I ) + (cid:13)(cid:13) |∇| p F z ( u + v ) (cid:13)(cid:13) L q pt L r dp (2 d − r sc + r x k w k X ( I ) . (cid:16) k u + v k pX ( I ) + (cid:13)(cid:13) |∇| p F z ( u + v ) (cid:13)(cid:13) L q pt L r dp (2 d − r sc + r x (cid:17) k w k X ( I ) . Thus, the claim will follow from (3.26), once we establish (cid:13)(cid:13) |∇| p F z ( u + v ) (cid:13)(cid:13) L q pt L r dp (2 d − r sc + r x . k u k p ( sc − sc X ( I ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) psc S ( I ) + k v k p ( sc − sc X ( I ) (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) psc S ( I ) . (3.32)For p ≥
1, this follows from Lemma 2.5 and (3.26): (cid:13)(cid:13) |∇| p F z ( u + v ) (cid:13)(cid:13) L q pt L r dp (2 d − r sc + r x . k u + v k p − X ( I ) k u + v k X ( I ) . k u + v k pX ( I ) . NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 19 To derive (3.32) for p <
1, we apply Lemma 2.6 (with s := p/ / < σ <
1) followedby H¨older’s inequality in the time variable, Sobolev embedding, and interpolation: (cid:13)(cid:13) |∇| p F z ( u + v ) (cid:13)(cid:13) L q pt L r dp (2 d − r sc + r x . k u + v k p − p σ X ( I ) (cid:13)(cid:13) |∇| σ ( u + v ) (cid:13)(cid:13) p σ L q t L r dd − r sc − σ ) x . (cid:13)(cid:13) |∇| σ ( u + v ) (cid:13)(cid:13) pL q t L r dd − r sc − σ ) x . k u k p − pσsc X ( I ) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) pσsc S ( I ) + k v k p − pσsc X ( I ) (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) pσsc S ( I ) . Invoking (3.26), this settles (3.32) and hence (3.30).To prove (3.31), we estimate (cid:13)(cid:13) |∇| s c [ F ( u + v ) − F ( u )] (cid:13)(cid:13) N ( I ) . (cid:13)(cid:13) |∇| s c − (cid:2) ∇ v · F ′ ( u + v ) (cid:3)(cid:13)(cid:13) N ( I ) + (cid:13)(cid:13) |∇| s c − (cid:2) ∇ u · (cid:0) F ′ ( u + v ) − F ′ ( u ) (cid:1)(cid:3)(cid:13)(cid:13) N ( I ) . (3.33)To estimate the first term on the right-hand side of (3.33), we use Lemmas 2.4 and 2.6together with H¨older’s inequality and interpolation: (cid:13)(cid:13) |∇| s c − (cid:2) ∇ v · F ′ ( u + v ) (cid:3)(cid:13)(cid:13) N ( I ) . (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) S ( I ) (cid:0) k u k pX ( I ) + k v k pX ( I ) (cid:1) + (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) sc S ( I ) k v k − sc X ( I ) k u + v k p − sc X ( I ) (cid:13)(cid:13) |∇| s c ( u + v ) (cid:13)(cid:13) − sc S ( I ) . (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) S ( I ) h k v k pX ( I ) + k u k p − sc X ( I ) k v k − sc X ( I ) + (cid:0) k u k p − sc X ( I ) + k v k p − sc X ( I ) (cid:1)(cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) − sc S ( I ) i . To estimate the second term on the right-hand side of (3.33), we use Lemma 2.8 togetherwith H¨older’s inequality, interpolation, and (3.26): (cid:13)(cid:13) |∇| s c − (cid:2) ∇ u · (cid:0) F ′ ( u + v ) − F ′ ( u ) (cid:1)(cid:3)(cid:13)(cid:13) N ( I ) . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) k v k pX ( I ) + (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) k u k sc − σ X ( I ) k v k p − sc − σ X ( I ) + (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) sc S ( I ) k u k − sc X ( I ) (cid:13)(cid:13) |∇| s c v (cid:13)(cid:13) − sc S ( I ) k v k p − sc X ( I ) , where s c − < σp < p . Denoting β := s c − σ and collecting all the estimates above we derive(3.31). (cid:3) We have now all the tools we need to attack Theorem 3.4. We start with the following:
Lemma 3.8 (Short-time perturbations) . Let d ≥ and assume the critical regularity s c satisfies (1.5) . Let I be a compact time interval containing zero and let ˜ u be an approximatesolution to (1.1) on I × R d in the sense that i ˜ u t = − ∆˜ u + F (˜ u ) + e for some function e . Assume that k ˜ u k L ∞ t ˙ H scx ( I × R d ) ≤ E for some positive constant E . Moreover, let u ∈ ˙ H s c x and assume that k ˜ u k X ( I ) ≤ δ (3.34) k u − ˜ u k ˙ H scx ≤ ε (3.35) (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I ) ≤ ε (3.36) for some small < δ = δ ( E ) and < ε < ε ( E ) . Then there exists a unique solution u : I × R d → C to (1.1) with initial data u at time t = 0 ; it satisfies k u − ˜ u k X ( I ) . ε (3.37) (cid:13)(cid:13) |∇| s c ( u − ˜ u ) (cid:13)(cid:13) S ( I ) . ε c ( d,p ) (3.38) (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) S ( I ) . E (3.39) k F ( u ) − F (˜ u ) k Y ( I ) . ε (3.40) (cid:13)(cid:13) |∇| s c (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ( I ) . ε c ( d,p ) , (3.41) for some positive constant c ( d, p ) .Proof. As explained at the beginning of the proof of Theorem 3.3, we may assume that u exists and merely show that it obeys the estimates stated above.We start by deriving some bounds on ˜ u and u . By Strichartz, Corollary 2.7, Lemma 3.6,(3.34), and (3.36), (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I ) . k ˜ u k L ∞ t ˙ H scx ( I × R d ) + (cid:13)(cid:13) |∇| s c F (˜ u ) (cid:13)(cid:13) N ( I ) + (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I ) . E + k ˜ u k pL p ( d +2)2 t,x ( I × R d ) (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I ) + ε . E + δ pθ (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) p (1 − θ ) S ( I ) + ε, where θ is as in Lemma 3.6. Choosing δ small depending on d, p, E and ε sufficiently smalldepending on E , we obtain (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I ) . E. (3.42)Moreover, by Lemma 3.5, Lemma 3.7, (3.34), and (3.36), (cid:13)(cid:13) e it ∆ ˜ u (cid:13)(cid:13) X ( I ) . k ˜ u k X ( I ) + k F (˜ u ) k Y ( I ) + (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I ) . δ + δ d +2 d − + ε . δ, provided δ and ε are chosen sufficiently small. Combining this with the triangle inequality,(3.26), the Strichartz inequality, and (3.35), we obtain (cid:13)(cid:13) e it ∆ u (cid:13)(cid:13) X ( I ) . (cid:13)(cid:13) e it ∆ ˜ u (cid:13)(cid:13) X ( I ) + k u − ˜ u k ˙ H scx . δ + ε . δ. Thus, another application of Lemma 3.5 combined with Lemma 3.7 gives k u k X ( I ) . (cid:13)(cid:13) e it ∆ u (cid:13)(cid:13) X ( I ) + k F ( u ) k Y ( I ) . δ + k u k d +2 d − X ( I ) . Choosing δ sufficiently small, the usual bootstrap argument yields k u k X ( I ) . δ. (3.43)Next we derive the claimed bounds on w := u − ˜ u . Note that w is a solution to ( iw t = − ∆ w + F (˜ u + w ) − F (˜ u ) − ew ( t ) = u − ˜ u . Using Lemma 3.5 together with Lemma 3.6, the Strichartz inequality, (3.35), and (3.36), wesee that k w k X ( I ) . k u − ˜ u k ˙ H scx + (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I ) + k F ( u ) − F (˜ u ) k Y ( I ) . ε + k F ( u ) − F (˜ u ) k Y ( I ) . To estimate the difference of the nonlinearities, we use Lemma 3.7, (3.34), (3.42): k F ( u ) − F (˜ u ) k Y ( I ) . h k ˜ u k p ( sc − sc X ( I ) (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) psc S ( I ) + k w k p ( sc − sc X ( I ) (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) psc S ( I ) i k w k X ( I ) . δ p ( sc − sc E psc k w k X ( I ) + (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) psc S ( I ) k w k p ( sc − sc X ( I ) . (3.44) NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 21 Thus, choosing δ sufficiently small depending only on E , we obtain k w k X ( I ) . ε + (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) psc S ( I ) k w k p ( sc − sc X ( I ) . (3.45)On the other hand, by the Strichartz inequality and the hypotheses, (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) S ( I ) . k u − ˜ u k ˙ H scx + (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I ) + (cid:13)(cid:13) |∇| s c (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ( I ) . ε + (cid:13)(cid:13) |∇| s c (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ( I ) . (3.46)To estimate the difference of the nonlinearities, we use (3.31) together with Lemma 3.6,(3.34), (3.42), and (3.43), (cid:13)(cid:13) |∇| s c (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ( I ) . (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) S ( I ) (cid:16) δ + δ p − sc E − sc (cid:17) (3.47)+ Eδ β k w k p − βX ( I ) + δ − sc E sc (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) − sc S ( I ) k w k p − sc X ( I ) for some 0 < β < p . Thus, choosing δ small depending only on E , (3.46) implies (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) S ( I ) . ε + k w k p − βX ( I ) + (cid:13)(cid:13) |∇| s c w (cid:13)(cid:13) − sc S ( I ) | w k p − sc X ( I ) . (3.48)Combining (3.45) with (3.48), the usual bootstrap argument yields (3.37) and (3.38),provided ε is chosen sufficiently small depending on E . By the triangle inequality, (3.38)and (3.42) imply (3.39).Claims (3.40) and (3.41) follow from (3.44) and (3.47) combined with (3.37) and (3.38),provided we take δ and ε sufficiently small depending on E . (cid:3) We are finally in a position to prove the stability result.
Proof of Theorem 3.4.
Our first goal is to show (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I ) ≤ C ( E, L ) . (3.49)Indeed, by (3.19) we may divide I into J = J ( L, η ) subintervals I j = [ t j , t j +1 ] such thaton each spacetime slab I j × R d k ˜ u k L p ( d +2)2 t,x ( I j × R d ) ≤ η for a small constant η > (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I j ) . k ˜ u ( t j ) k ˙ H scx + (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ( I j ) + (cid:13)(cid:13) |∇| s c F (˜ u ) (cid:13)(cid:13) N ( I j ) . E + ε + k ˜ u k pL p ( d +2)2 t,x ( I j × R d ) (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I j ) . E + ε + η p (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I j ) . Thus, choosing η > d, p and ε sufficiently small depending on E , weobtain (cid:13)(cid:13) |∇| s c ˜ u (cid:13)(cid:13) S ( I j ) . E. Summing this over all subintervals I j , we derive (3.49).Using Lemma 3.6 together with (3.49), we obtain k ˜ u k X ( I ) ≤ C ( E, L ) (3.50)By (3.50), we may divide I into J = J ( E, L ) subintervals I j = [ t j , t j +1 ] such that on eachspacetime slab I j × R d k ˜ u k X ( I j ) ≤ δ for some small δ = δ ( E ) > Thus, choosing ε sufficiently small (depending on J and E ), we may apply Lemma 3.8to obtain for each 0 ≤ j < J and all 0 < ε < ε , k u − ˜ u k X ( I j ) ≤ C ( j ) ε c ( d,p ) j (cid:13)(cid:13) |∇| s c ( u − ˜ u ) (cid:13)(cid:13) S ( I j ) ≤ C ( j ) ε c ( d,p ) j +1 (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) ˙ S ( I j ) ≤ C ( j ) E k F ( u ) − F (˜ u )] k Y ( I j ) ≤ C ( j ) ε c ( d,p ) j (cid:13)(cid:13) |∇| s c (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ( I j ) ≤ C ( j ) ε c ( d,p ) j +1 , (3.51)provided we can show k u ( t j ) − ˜ u ( t j ) k ˙ H sc . ε c ( d,p ) j ≤ ε (3.52)for each 0 ≤ j < J , where ε is as in Lemma 3.8. By the Strichartz inequality and theinductive hypothesis, k u ( t j ) − ˜ u ( t j ) k ˙ H scx . k u − ˜ u k ˙ H scx + (cid:13)(cid:13) |∇| s c e (cid:13)(cid:13) N ([0 ,t j ]) + (cid:13)(cid:13) |∇| s c (cid:2) F ( u ) − F (˜ u ) (cid:3)(cid:13)(cid:13) N ([0 ,t j ]) . ε + j − X k =0 C ( k ) ε c ( d,p ) k +1 . Taking ε sufficiently small compared to ε , we see that (3.52) is satisfied.Summing the bounds in (3.51) over all subintervals I j and using Lemma 3.6, we derive(3.22) through (3.24). This completes the proof of the theorem. (cid:3) Reduction to almost periodic solutions
The goal of this section is to prove Theorem 1.12. In order to achieve this, we repeat theargument presented in [19]. Since the procedure is by now standard, we content ourselveswith identifying the main steps and indicating, whenever necessary, the changes that appearwith respect to the presentation in [19]. See also [13, 14, 15, 20, 34] for similar argumentsin other contexts.We start by presenting the setup. Throughout this section, we fix a dimension d ≥ s c = d − or it satisfies (1.5). For any0 ≤ E < ∞ , we define L ( E ) := sup { S I ( u ) : u : I × R d → C such that sup t ∈ I (cid:13)(cid:13) |∇| s c u ( t ) (cid:13)(cid:13) ≤ E } , where the supremum is taken over all solutions u : I × R d → C to (1.1) obeying (cid:13)(cid:13) |∇| s c u ( t ) (cid:13)(cid:13) ≤ E . Thus, L : (cid:2) , ∞ ) → [0 , ∞ ] is a non-decreasing function. Moreover, from Theorem 1.6, L ( E ) . d E p ( d +2)4 for E ≤ η , where η = η ( d, p ) is the threshold from the small data theory.From Theorems 3.3 and 3.4, we see that L is continuous in both settings considered here.Therefore, there must exist a unique critical E c ∈ (0 , ∞ ] such that L ( E ) < ∞ for E < E c and L ( E ) = ∞ for E ≥ E c . In particular, if u : I × R d → C is a maximal-lifespan solutionto (1.1) such that sup t ∈ I (cid:13)(cid:13) |∇| s c u ( t ) (cid:13)(cid:13) < E c , then u is global and moreover, S R ( u ) ≤ L (cid:0) k u k L ∞ t ˙ H scx (cid:1) . Failure of Theorems 1.4 or 1.5 is equivalent to 0 < E c < ∞ .Following the presentation in [19, 34], the main step in proving Theorem 1.12 is to provea Palais–Smale condition modulo the symmetries of the equation. With the Palais–Smalecondition in place, the proof of Theorem 1.12 is standard; see, for example, [13, 14, 19, 34]. NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 23 Proposition 4.1 (Palais-Smale condition modulo symmetries) . Assume we are either inthe setting of Theorem 1.4 or Theorem 1.5. Let u n : I n × R d C be a sequence of solutionsto (1.1) such that lim sup n →∞ sup t ∈ I n (cid:13)(cid:13) |∇| s c u n ( t ) (cid:13)(cid:13) = E c (4.1) and let t n ∈ I n be a sequence of times such that lim n →∞ S ≥ t n ( u n ) = lim n →∞ S ≤ t n ( u n ) = ∞ . Then the sequence u n ( t n ) has a subsequence which converges in ˙ H s c x ( R d ) modulo symmetries. The proof of this proposition follows the same recipe as that used to prove the anal-ogous statement in [19]. The main ingredients are the linear profile decomposition fromLemma 2.12 and the stability results Theorems 3.3 and 3.4. The only new difficulty appearswhen one endeavors to prove decoupling of the nonlinear profiles. In the energy-criticalsetting, one uses the pointwise inequality (cid:12)(cid:12)(cid:12) ∇ (cid:16) J X j =1 F ( f j ) − F (cid:0) J X j =1 f j (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) . J X j = j ′ |∇ f j || f j ′ | d − , which does not carry over to the case of a non-integer number of derivatives. Let us firstdiscuss the case 0 < p ≤
1; in this scenario, we use (2.6) to obtain the following substitutefor the inequality above: D s c − (cid:18) ∇ h J X j =1 F ( f j ) − F (cid:0) J X j =1 f j (cid:1)i(cid:19) . D s c − (cid:18) J X j =1 ∇ f j h F ′ ( f j ) − F ′ (cid:0) J X j ′ =1 f j ′ (cid:1)i(cid:19) . J X j = j ′ (cid:26) D s c − (cid:0) ∇ f j (cid:1) | f j ′ | p + (cid:20) J X l =1 D σ ( f l ) (cid:21) sσ M (cid:0) |∇ f j | − p (cid:1) − p M (cid:0) f j ′ (cid:1) p − sσ (cid:27) for some s c − < σp < p . We remind the reader that M denotes the Hardy–Littlewoodmaximal function, which commutes with the symmetries of the equation. The operator D s is defined in (2.5) and it behaves like |∇| s under symmetries.In the case discussed above, one has both a small-power non-polynomial nonlinearityand a non-integer number of derivatives 1 < s c <
2, which makes it the most awkward ofthe scenarios we need to consider. The remaining cases of Theorems 1.4 and 1.5, can behandled using various permutations of the techniques discussed above or some alternatives.In particular, we draw the reader’s attention to [14] which considers the cubic nonlinearitywith s c = . 5. The finite-time blowup solution
In this section we preclude scenario I described in Theorem 1.13. We start by consideringfinite-time blowup solutions in the setting of Theorem 1.5.
Theorem 5.1 (Absence of finite-time blowup solutions) . Let d ≥ and assume the criticalregularity s c satisfies (1.5) . Then there are no finite-time blowup solutions to (1.1) in thesense of Theorem 1.13.Proof. We argue by contradiction. Assume that there exists a solution u : I × R d → C thatis a finite-time blowup solution in the sense of Theorem 1.13. Assume also, without loss ofgenerality, that the solution u blows up in finite time in the future, that is, T := sup I < ∞ . By hypothesis and Sobolev embedding, k u k L ∞ t L dp x ( I × R d ) . (cid:13)(cid:13) |∇| s c − u (cid:13)(cid:13) L ∞ t L dd − x ( I × R d ) . k u k L ∞ t ˙ H scx ( I × R d ) . u . Thus, using the Duhamel formula (1.7) into the future together with the Strichartz andH¨older inequalities, as well as the fractional chain rule, we obtain (cid:13)(cid:13) |∇| s c − u ( t ) (cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)Z Tt e i ( t − s )∆ |∇| s c − F ( u ( s )) ds (cid:13)(cid:13)(cid:13) . (cid:13)(cid:13) |∇| s c − F ( u ) (cid:13)(cid:13) L t L dd +2 x ([ t,T ) × R d ) . ( T − t ) (cid:13)(cid:13) |∇| s c − u (cid:13)(cid:13) L ∞ t L dd − x ([ t,T ) × R d ) k u k pL ∞ t L dp x ([ t,T ) × R d ) . u ( T − t ) . Interpolating with u ∈ L ∞ t ˙ H s c x (recalling that 1 < s c < E ( u ) → t → T . Invoking the conservation of energy, we deduce that u ≡ u is a blowup solution. (cid:3) We consider next finite-time blowup solutions in the setting of Theorem 1.4.
Theorem 5.2 (Absence of finite-time blowup solutions – the cubic) . Let d ≥ and assume p = 2 . Then there are no finite-time blowup solutions to (1.1) in the sense of Theorem 1.13.Proof. Again, we argue by contradiction. Let u : I × R d → C be a finite-time blowupsolution in the sense of Theorem 1.13 and assume that T := sup I < ∞ .By Sobolev embedding and the hypothesis, k u k L ∞ t L dx ( I × R d ) . (cid:13)(cid:13) |∇| d − u (cid:13)(cid:13) L ∞ t L dd − x ( I × R d ) . k u k L ∞ t ˙ H d − x ( I × R d ) . u . Thus, using the Duhamel formula (1.7) into the future together with the Strichartz andH¨older inequalities, as well as the fractional chain rule, we obtain (cid:13)(cid:13) |∇| d − u ( t ) (cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)Z Tt e i ( t − s )∆ |∇| d − F ( u ( s )) ds (cid:13)(cid:13)(cid:13) . (cid:13)(cid:13) |∇| d − F ( u ) (cid:13)(cid:13) L t L dd +2 x ([ t,T ) × R d ) . ( T − t ) (cid:13)(cid:13) |∇| d − u (cid:13)(cid:13) L ∞ t L dd − x ([ t,T ) × R d ) k u k L ∞ t L dx ([ t,T ) × R d ) . u ( T − t ) . (5.1)In particular, u ( t ) ∈ ˙ H d − x .In the case when d = 5, one can interpolate between (5.1) and the hypothesis u ∈ L ∞ t ˙ H d − x ( I × R d ) to derive that k∇ u ( t ) k → t → T , and hence, by Sobolev embedding,the energy E ( u ( t )) → t → T . Using the conservation of energy, we deduce that u ≡ S I ( u ) = ∞ .To handle higher dimensions, we iterate the computations in (5.1) with one less derivativeto deduce that (cid:13)(cid:13) |∇| d − u ( t ) (cid:13)(cid:13) . u ( T − t ) . For d = 6 this immediately implies that u must have zero mass, while for d = 7, theargument used to handle dimension d = 5 implies that u must have zero energy. In bothcases, we derive a contradiction to the fact that u is a blowup solution.To derive a contradiction for dimensions d ≥
8, we iterate the argument presented above.This finishes the proof of the theorem. (cid:3)
NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 25 Negative regularity
In this section we prove that in scenarios II and III described in Theorem 1.13, thesolution u admits negative regularity; more precisely, it lies in L ∞ t ˙ H − εx for some ε >
0. Inparticular, this shows that the solution decays sufficiently rapidly (in space) to belong to L ∞ t L x . We first consider the setting of Theorem 1.5. At the end of this section we explainthe changes needed to prove negative regularity in the setting of Theorem 1.4. Theorem 6.1 (Negative regularity for scenarios II and III) . Let d ≥ and assume thecritical regularity s c obeys (1.5) . Let u be a global solution to (1.1) that is almost periodicmodulo symmetries. Suppose also that sup t ∈ I (cid:13)(cid:13) |∇| s c u ( t ) (cid:13)(cid:13) L x < ∞ (6.1) and inf t ∈ I N ( t ) ≥ . (6.2) Then u ∈ L ∞ t ˙ H − εx for some ε = ε ( d ) > . In particular, u ∈ L ∞ t L x . To prove Theorem 6.1, we employ the argument used in [19] to treat the energy-criticalcase, s c = 1. We achieve our goal in two steps: First, we ‘break’ scaling in a Lebesque space;more precisely, we prove that our solution lives in L ∞ t L qx for some 2 < q < dp . Next, weuse a double Duhamel trick to upgrade this to u ∈ L ∞ t ˙ H s c − s x for some s = s ( d, p, q ) > u be a solution to (1.1) that obeys the hypotheses of Theorem 6.1. Let η > N = N ( η ) such that (cid:13)(cid:13) |∇| s c u ≤ N (cid:13)(cid:13) L ∞ t L x ≤ η. (6.3)We turn now to our first step, that is, breaking scaling in a Lebesgue space. To this end,we define A ( N ) := N d − − p sup t ∈ R k u N ( t ) k L dd − x for d = 5 , N d − − p − p sup t ∈ R k u N ( t ) k L dd − − px for d ≥ N ≤ N . To simplify the formulas appearing below, we introduce thenotation α ( d ) := ( − d − + p for d = 5 , − d − + p + p for d ≥ . Note that by (1.5), 0 < α ( d ) < d = 5 , < α ( d ) < p for d ≥
7. Note also that byBernstein’s inequality and (6.1), A ( N ) . N d − p k u N k L ∞ t L x . (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) L ∞ t L x < ∞ . (6.4)We next prove a recurrence formula for A ( N ). Lemma 6.2 (Recurrence) . For all N ≤ N , A ( N ) . u (cid:0) NN (cid:1) min { ,p }− α ( d ) + η p X N ≤ N ≤ N (cid:0) NN (cid:1) min { ,p }− α ( d ) A ( N ) + η p X N < N (cid:0) N N (cid:1) α ( d ) A ( N ) . Proof.
We first give the proof in dimensions d ≥
7. Once this is completed, we will explainthe changes necessary to treat d = 5 , Fix N ≤ N . By time-translation symmetry, it suffices to prove N − α ( d ) k u N (0) k L dd − − px . u (cid:0) NN (cid:1) p − α ( d ) + η p X N ≤ N ≤ N (cid:0) NN (cid:1) p − α ( d ) A ( N )+ η p X N < N (cid:0) N N (cid:1) α ( d ) A ( N ) . (6.5)Using the Duhamel formula (1.7) into the future followed by the triangle inequality,Bernstein, and the dispersive inequality, we estimate N − α ( d ) k u N (0) k L dd − − px ≤ N − α ( d ) Z N − (cid:13)(cid:13) e − it ∆ P N F ( u ( t )) (cid:13)(cid:13) L dd − − px dt + N − α ( d ) Z ∞ N − (cid:13)(cid:13) e − it ∆ P N F ( u ( t )) (cid:13)(cid:13) L dd − − px dt . N p +1 − α ( d ) Z N − (cid:13)(cid:13) e − it ∆ P N F ( u ( t )) (cid:13)(cid:13) L x dt + N − α ( d ) k P N F ( u ) k L ∞ t L dd +2+ px Z ∞ N − t − p dt . N p − − α ( d ) k P N F ( u ) k L ∞ t L x + N p − α ( d ) k P N F ( u ) k L ∞ t L dd +2+ px . N p − α ( d ) k P N F ( u ) k L ∞ t L dd +2+ px . (6.6)Using the Fundamental Theorem of Calculus, we decompose F ( u ) = O ( | u >N || u ≤ N | p ) + O ( | u >N | p ) + F ( u N ≤·≤ N )+ u < N Z F z (cid:0) u N ≤·≤ N + θu < N (cid:1) dθ (6.7)+ u < N Z F ¯ z (cid:0) u N ≤·≤ N + θu < N (cid:1) dθ. The contribution to the right-hand side of (6.6) coming from terms that contain at leastone copy of u >N can be estimated in the following manner: Using H¨older, Bernstein,Sobolev embedding, and (6.1), N p − α ( d ) k P N O ( | u >N || u | p ) (cid:13)(cid:13) L ∞ t L dd +2+ px . N p − α ( d ) k u k pL ∞ t L dp x k u >N k L ∞ t L dd − px . u N p − α ( d ) N α ( d ) − p . (6.8)Thus, this contribution is acceptable.Next we turn to the contribution to the right-hand side of (6.6) coming from the last twoterms in (6.7); it suffices to consider the first of them since similar arguments can be usedto deal with the second.First we note that as |∇| s c u ∈ L ∞ t L x , by Sobolev embedding we must have ∇ u ∈ L ∞ t L dp p x . Thus, an application of Lemma 2.9 together with (6.1) yield (cid:13)(cid:13) P > N F z ( u ) (cid:13)(cid:13) L ∞ t L d px . N − p k∇ u k pL ∞ t L dp px . N − p (cid:13)(cid:13) |∇| s c u (cid:13)(cid:13) pL ∞ t L x . u N − p . Thus, by H¨older’s inequality and (6.3), N p − α ( d ) (cid:13)(cid:13)(cid:13) P N (cid:16) u < N Z F z (cid:0) u N ≤·≤ N + θu < N (cid:1) dθ (cid:17)(cid:13)(cid:13)(cid:13) L ∞ t L dd +2+ px . N p − α ( d ) k u < N k L ∞ t L dd − − px (cid:13)(cid:13)(cid:13) P > N (cid:16)Z F z (cid:0) u N ≤·≤ N + θu < N (cid:1) dθ (cid:17)(cid:13)(cid:13)(cid:13) L ∞ t L d px NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 27 . N − α ( d ) k u < N k L ∞ t L dd − − px (cid:13)(cid:13) |∇| s c u ≤ N (cid:13)(cid:13) pL ∞ t L x . u η p X N < N (cid:0) N N (cid:1) α ( d ) A ( N ) . (6.9)Hence, the contribution coming from the last two terms in (6.7) is acceptable.We are left to estimate the contribution of F ( u N ≤·≤ N ) to the right-hand side of (6.6).We need only show k F ( u N ≤·≤ N ) k L ∞ t L dd +2+ px . u η p X N ≤ N ≤ N N α ( d ) − p A ( N ) . (6.10)As d ≥ s c satisfies (1.5), we must have p <
1. Using the triangleinequality, Bernstein, (6.3), and H¨older, we estimate k F ( u N ≤·≤ N ) k L ∞ t L dd +2+ px . X N ≤ N ≤ N (cid:13)(cid:13) u N | u N ≤·≤ N | p (cid:13)(cid:13) L ∞ t L dd +2+ px . X N ≤ N ,N ≤ N (cid:13)(cid:13) u N | u N | p (cid:13)(cid:13) L ∞ t L dd +2+ px . X N ≤ N ≤ N ≤ N k u N k L ∞ t L dd − − px k u N k pL ∞ t L dp px + X N ≤ N ≤ N ≤ N k u N k pL ∞ t L dp px k u N k − pL ∞ t L dd − − px k u N k pL ∞ t L dd − − px . X N ≤ N ≤ N ≤ N k u N k L ∞ t L dd − − px N − p (cid:13)(cid:13) |∇| s c u N (cid:13)(cid:13) pL ∞ t L x + X N ≤ N ≤ N ≤ N N − p (cid:13)(cid:13) |∇| s c u N (cid:13)(cid:13) pL ∞ t L x k u N k − pL ∞ t L dd − − px k u N k pL ∞ t L dd − − px . u η p X N ≤ N ≤ N N α ( d ) − p A ( N )+ η p X N ≤ N ≤ N ≤ N (cid:0) N N (cid:1) p (cid:2) N α ( d ) − p A ( N ) (cid:3) − p (cid:2) N α ( d ) − p A ( N ) (cid:3) p . u η p X N ≤ N ≤ N N α ( d ) − p A ( N ) . This proves (6.10) and so completes the proof of the lemma in dimensions d ≥ d = 5 ,
6. Note that in this case, our assumptions guarantee that 1 < p < N − α ( d ) k u N (0) k L dd − x . N − α ( d ) k P N F ( u ) k L ∞ t L dd +3 x , which we estimate by decomposing the nonlinearity as in (6.7). The analogue of (6.8) inthis case is N − α ( d ) k P N O ( | u >N || u | p ) (cid:13)(cid:13) L ∞ t L dd +3 x . N − α ( d ) k u k pL ∞ t L dp x k u >N k L ∞ t L dd − x . u (cid:0) NN (cid:1) − α ( d ) . Using Bernstein and Lemma 2.5 together with (6.3) and Sobolev embedding, we replace(6.9) by N − α ( d ) (cid:13)(cid:13)(cid:13) P N (cid:16) u < N Z F z (cid:0) u N ≤·≤ N + θu < N (cid:1) dθ (cid:17)(cid:13)(cid:13)(cid:13) L ∞ t L dd +3 x . N − α ( d ) k u < N k L ∞ t L dd − x (cid:13)(cid:13)(cid:13) P > N (cid:16)Z F z (cid:0) u N ≤·≤ N + θu < N (cid:1) dθ (cid:17)(cid:13)(cid:13)(cid:13) L ∞ t L d x . N − α ( d ) k u < N k L ∞ t L dd − x k∇ u ≤ N k L ∞ t L dp px k u ≤ N k p − L ∞ t L dp x . X N < N (cid:0) N N (cid:1) α ( d ) A ( N ) (cid:13)(cid:13) |∇| s c u ≤ N (cid:13)(cid:13) pL ∞ t L x . u η p X N < N (cid:0) N N (cid:1) α ( d ) A ( N ) . Finally, arguing as for (6.10), we estimate k F ( u N ≤·≤ N ) k L ∞ t L dd +3 x . X N ≤ N ,N ≤ N (cid:13)(cid:13) u N u N | u N ≤·≤ N | p − (cid:13)(cid:13) L ∞ t L dd +3 x . X N ≤ N ≤ N ,N ≤ N k u N k L ∞ t L dd − x k u N k L ∞ t L dp x k u N k p − L ∞ t L dp x + X N ≤ N ≤ N ≤ N ≤ N k u N k − pL ∞ t L dd − x k u N k p − L ∞ t L dp x k u N k L ∞ t L dp x k u N k p − L ∞ t L dd − x . u X N ≤ N ≤ N ,N ≤ N k u N k L ∞ t L dd − x ηN − p (cid:0) ηN − p (cid:1) p − + X N ≤ N ≤ N ≤ N ≤ N k u N k − pL ∞ t L dd − x (cid:0) ηN − p (cid:1) p − ηN − p k u N k p − L ∞ t L dd − x . u η p X N ≤ N ≤ N N α ( d ) − A ( N )+ η p X N ≤ N ≤ N ≤ N (cid:0) N N (cid:1) p − (cid:2) N α ( d ) − A ( N ) (cid:3) − p (cid:2) N α ( d ) − A ( N ) (cid:3) p − . u η p X N ≤ N ≤ N N α ( d ) − A ( N ) . Putting everything together completes the proof of the lemma when d = 5 , (cid:3) This lemma leads very quickly to our first goal:
Proposition 6.3 ( L p breach of scaling) . Let u be as in Theorem 6.1. Then u ∈ L ∞ t L qx for some < q < dp . (6.11) In particular, for ≤ s ≤ s c , |∇| s u ∈ L ∞ t L x = ⇒ |∇| s F ( u ) ∈ L ∞ t L rx for some r < dd +4 . (6.12) Proof.
We first consider (6.11). We will only present the details for d ≥
7. The treatmentof d = 5 , NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 29 Combining Lemma 6.2 with Lemma 2.13, we deduce k u N k L ∞ t L dd − − px . u N p − for all N ≤ N . (6.13)In applying Lemma 2.13, we set N = 10 · − k N , x k = A (10 · − k N ), and take η sufficientlysmall. Note that x k ∈ ℓ ∞ by virtue of (6.4).By interpolation followed by (6.13), Bernstein, and (6.1), k u N k L ∞ t L qx . k u N k d ( q − q (2+ p ) L ∞ t L dd − − px k u N k d − q ( d − − p ) q (2+ p ) L ∞ t L x . u N dp ( q − q (2+ p ) − N − ( d − p ) d − q ( d − − p ) q (2+ p ) for all N ≤ N . Note that the power of N appearing in the formula above is positive for d (2 p + dp − dp +( d − d +4) p − d − < q < dp . Thus, for these values of q , Bernstein together with (6.1) yield k u k L ∞ t L qx ≤ k u ≤ N k L ∞ t L qx + k u >N k L ∞ t L qx . u X N ≤ N N + X N>N N p − dq . u , which completes the proof of (6.11).We turn now to (6.12). For 0 ≤ s ≤
1, we use Lemma 2.5 to estimate (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L rx . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) L ∞ t L x k u k pL ∞ t L rp − rx . We will return to this shortly, but first we consider the case 1 < s ≤ s c . Using Lemma 2.4together with Sobolev embedding, (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L rx . (cid:13)(cid:13) |∇| s − (cid:0) ∇ u · F ′ ( u ) (cid:1)(cid:13)(cid:13) L ∞ t L rx . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) L ∞ t L x k u k pL ∞ t L rp − rx + k∇ u k L ∞ t L dp px (cid:13)(cid:13) |∇| s − F ′ ( u ) (cid:13)(cid:13) L ∞ t L dprdp − r (2+ p ) x . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) L ∞ t L x (cid:16) k u k pL ∞ t L rp − rx + (cid:13)(cid:13) |∇| s − F ′ ( u ) (cid:13)(cid:13) L ∞ t L dprdp − r (2+ p ) x (cid:17) . For p ≥
1, we invoke Lemma 2.5 and then use Sobolev embedding to estimate (cid:13)(cid:13) |∇| s − F ′ ( u ) (cid:13)(cid:13) L ∞ t L dprdp − r (2+ p ) x . (cid:13)(cid:13) |∇| s − u (cid:13)(cid:13) L ∞ t L dd − x k u k p − L ∞ t L dpr ( p − dp − dpr − rx . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) L ∞ t L x k u k p − L ∞ t L dpr ( p − dp − dpr − rx . If instead p <
1, we use Lemma 2.6 and Sobolev embedding to obtain (cid:13)(cid:13) |∇| s − F ′ ( u ) (cid:13)(cid:13) L ∞ t L dprdp − r (2+ p ) x . (cid:13)(cid:13) |∇| σ u (cid:13)(cid:13) s − σ L ∞ t L dp σpx k u k p − s − σ L ∞ t L dpr ( σp +1 − s ) dpσ − σprs − r ( s − − σrx . (cid:13)(cid:13) |∇| s u (cid:13)(cid:13) s − σ L ∞ t L x k u k p − s − σ L ∞ t L dpr ( σp +1 − s ) dpσ − σprs − r ( s − − σrx for some s − p < σ < ≤ s ≤ s c , to obtain (6.12) it thus suffices to note that for r < dd +4 the exponents rp − r , dpr ( p − dp − dpr − r , dpr ( σp +1 − s ) dpσ − σprs − r ( s − − σr are all less than dp and converge to this number as r → dd +4 . Thus, (6.11) yields the claim. (cid:3) Following [19], our second step is to use the double Duhamel trick to upgrade (6.11) to anegative regularity statement in L x -based Sobolev spaces. The arguments we present followclosely the ones in [19]. For the sake of completeness, we present the details below. Proposition 6.4 (Some negative regularity) . Let d ≥ and assume the critical regularity s c obeys (1.5) . Let u be as in Theorem 6.1. Assume further that |∇| s F ( u ) ∈ L ∞ t L rx for some r < dd +4 and some ≤ s ≤ s c . Then there exists s = s ( d, r ) > such that u ∈ L ∞ t ˙ H s − s + x .Proof. The proposition will follow once we establish (cid:13)(cid:13) |∇| s u N (cid:13)(cid:13) L ∞ t L x . u N s for all N > s := dr − d +42 > . (6.14)Indeed, by Bernstein combined with (6.1), (cid:13)(cid:13) |∇| s − s + u (cid:13)(cid:13) L ∞ t L x ≤ (cid:13)(cid:13) |∇| s − s + u ≤ (cid:13)(cid:13) L ∞ t L x + (cid:13)(cid:13) |∇| s − s + u > (cid:13)(cid:13) L ∞ t L x . u X N ≤ N + X N> N ( s − s +) − s c . u . Thus, we are left to prove (6.14). By time-translation symmetry, it suffices to prove (cid:13)(cid:13) |∇| s u N (0) (cid:13)(cid:13) L x . u N s for all N > s := dr − d +42 > . (6.15)Using the Duhamel formula (1.7) both in the future and in the past, we write (cid:13)(cid:13) |∇| s u N (0) (cid:13)(cid:13) L x = lim T →∞ lim T ′ →−∞ (cid:10) i Z T e − it ∆ P N |∇| s F ( u ( t )) dt, − i Z T ′ e − iτ ∆ P N |∇| s F ( u ( τ )) dτ (cid:11) ≤ Z ∞ Z −∞ (cid:12)(cid:12)(cid:12)(cid:10) P N |∇| s F ( u ( t )) , e i ( t − τ )∆ P N |∇| s F ( u ( τ )) (cid:11)(cid:12)(cid:12)(cid:12) dt dτ. We estimate the term inside the integrals in two ways. On one hand, using H¨older and thedispersive estimate, (cid:12)(cid:12)(cid:12)(cid:10) P N |∇| s F ( u ( t )) , e i ( t − τ )∆ P N |∇| s F ( u ( τ )) (cid:11)(cid:12)(cid:12)(cid:12) . (cid:13)(cid:13) P N |∇| s F ( u ( t )) (cid:13)(cid:13) L rx (cid:13)(cid:13) e i ( t − τ )∆ P N |∇| s F ( u ( τ )) (cid:13)(cid:13) L r ′ x . | t − τ | d − dr (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L rx . On the other hand, using Bernstein, (cid:12)(cid:12)(cid:12)(cid:10) P N |∇| s F ( u ( t )) , e i ( t − τ )∆ P N |∇| s F ( u ( τ )) (cid:11)(cid:12)(cid:12)(cid:12) . (cid:13)(cid:13) P N |∇| s F ( u ( t )) (cid:13)(cid:13) L x (cid:13)(cid:13) e i ( t − τ )∆ P N |∇| s F ( u ( τ )) (cid:13)(cid:13) L x . N d − dr ) (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L rx . Thus, (cid:13)(cid:13) |∇| s u N (0) (cid:13)(cid:13) L x . (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L rx Z ∞ Z −∞ min {| t − τ | − , N } dr − d dt dτ . N s (cid:13)(cid:13) |∇| s F ( u ) (cid:13)(cid:13) L ∞ t L rx . To obtain the last inequality we used the fact that dr − d > r < dd +4 . Thus (6.15)holds; this finishes the proof of the proposition. (cid:3) The proof of Theorem 6.1 will follow from iterating Proposition 6.4 finitely many times.
Proof of Theorem 6.1.
Proposition 6.3 allows us to apply Proposition 6.4 with s = s c . Weconclude that u ∈ L ∞ t ˙ H s c − s + x for some s = s ( d, r ) >
0. Combining this with (6.12), wededuce that |∇| s c − s + F ( u ) ∈ L ∞ t L rx for some r < dd +4 . We are thus in the position to apply NERGY-SUPERCRITICAL NLS: CRITICAL ˙ H s -BOUNDS IMPLY SCATTERING 31 Proposition 6.4 again and obtain u ∈ L ∞ t ˙ H s c − s + x . Iterating this procedure finitely manytimes, we derive u ∈ L ∞ t ˙ H − εx for any 0 < ε < s .This completes the proof of Theorem 6.1. (cid:3) To prove the equivalent of Theorem 6.1 in the setting of Theorem 1.4, one repeats thearguments presented above with A ( N ) := N d − sup t ∈ R k u N ( t ) k L dd − x . The low-to-high frequency cascade
In this section, we use the negative regularity proved in the previous section to precludelow-to-high frequency cascade solutions.
Theorem 7.1 (Absence of cascades) . In the settings of Theorems 1.4 and 1.5 there are nosolutions to (1.1) that are low-to-high frequency cascades in the sense of Theorem 1.13.Proof.
Suppose for a contradiction that there existed such a solution u . Then, by the resultsproved in Section 6, u ∈ L ∞ t L x ; thus, by the conservation of mass,0 ≤ M ( u ) = M ( u ( t )) = Z R d | u ( t, x ) | dx < ∞ for all t ∈ R . Fix t ∈ R and let η > Z | ξ |≤ c ( η ) N ( t ) | ξ | s c | ˆ u ( t, ξ ) | dξ ≤ η. On the other hand, as u ∈ L ∞ t ˙ H − εx for some ε > Z | ξ |≤ c ( η ) N ( t ) | ξ | − ε | ˆ u ( t, ξ ) | dξ . u . Hence, by H¨older’s inequality, Z | ξ |≤ c ( η ) N ( t ) | ˆ u ( t, ξ ) | dξ . u η εsc + ε . (7.1)Meanwhile, by elementary considerations and u ∈ L ∞ t ˙ H s c x , Z | ξ |≥ c ( η ) N ( t ) | ˆ u ( t, ξ ) | dξ ≤ [ c ( η ) N ( t )] − s c Z R d | ξ | s c | ˆ u ( t, ξ ) | dξ ≤ [ c ( η ) N ( t )] − s c (cid:13)(cid:13) |∇| s c u ( t ) (cid:13)(cid:13) . u [ c ( η ) N ( t )] − s c . (7.2)Collecting (7.1) and (7.2) and using Plancherel’s theorem, we obtain0 ≤ M ( u ) . u c ( η ) − s c N ( t ) − s c + η εsc + ε for all t ∈ R . As u is a low-to-high cascade, there is a sequence of times t n → ∞ so that N ( t n ) → ∞ . As η > M ( u ) = 0 and hence u is identicallyzero. This contradicts the fact that S R ( u ) = ∞ , thus settling Theorem 7.1. (cid:3) The soliton
In this section, we use the negative regularity proved in Section 6 to preclude soliton-likesolutions.
Theorem 8.1 (Absence of solitons) . In the settings of Theorems 1.4 and 1.5 there are noglobal solutions to (1.1) that are solitons in the sense of Theorem 1.13.Proof.
As usual, we will use a monotonicity formula to preclude soliton-like solutions. Sincewe are in the defocusing case, we will use the interaction Morawetz inequality introduced in[7]. For the high dimensional case discussed here, the details of this derivation can be foundin [33] or [37].To prove Theorem 8.1, we argue by contradiction. We assume there exists a solution u to (1.1) which is a soliton in the sense of Theorem 1.13. Then, by the negative regularityresults proved in Section 6, u ∈ L ∞ t H x . The interaction Morawetz inequality yields Z I Z R d Z R d | u ( t, x ) | | u ( t, y ) | | x − y | dx dy dt . k u k L ∞ t L x ( I × R d ) k∇ u k L ∞ t L x ( I × R d ) . u I ⊂ R . As in dimension d convolution with | x | − is basi-cally the same as the fractional integration operator |∇| − ( d − , the interaction Morawetzinequality yields (cid:13)(cid:13) |∇| − d − ( | u | ) (cid:13)(cid:13) L t,x ( I × R d ) . u . By [37, Lemma 5.6], this implies (cid:13)(cid:13) |∇| − d − u (cid:13)(cid:13) L t,x ( I × R d ) . u . Interpolating between this estimate and the fact that u ∈ L ∞ t ˙ H x , we derive k u k L d +1 t L d +1) d − x ( I × R d ) . u I ⊂ R .Next we claim that k u ( t ) k L d +1) d − x & u t ∈ R . (8.2)Otherwise, there exists a time sequence t n such that u ( t n ) converges weakly to zero in L d +1) d − x . As u ( t ) is uniformly bounded in ˙ H s c x , this implies that u ( t n ) converges weakly tozero in ˙ H s c x . As the orbit of u is precompact in ˙ H s c x and u is not identically zero, we derivea contradiction.Using (8.1) and (8.2), we easily derive a contradiction by taking the interval I to besufficiently long. (cid:3) References [1] P. Begout and A. Vargas,
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