aa r X i v : . [ m a t h . A P ] S e p THE CUBIC FOURTH-ORDER SCHR ¨ODINGER EQUATION
BENOIT PAUSADER
Abstract.
Fourth-order Schr¨odinger equations have been introduced by Karp-man and Shagalov to take into account the role of small fourth-order dispersionterms in the propagation of intense laser beams in a bulk medium with Kerrnonlinearity. In this paper we investigate the cubic defocusing fourth orderSchr¨odinger equation i∂ t u + ∆ u + | u | u = 0in arbitrary space dimension R n for arbitrary initial data. We prove that theequation is globally well-posed when n ≤ n ≥
9, with theadditional important information that scattering holds true when 5 ≤ n ≤ Introduction
Fourth-order Schr¨odinger equations have been introduced by Karpman [15] andKarpman and Shagalov [16] to take into account the role of small fourth-order dis-persion terms in the propagation of intense laser beams in a bulk medium withKerr nonlinearity. Such fourth-order Schr¨odinger equations have been studied fromthe mathematical viewpoint in Fibich, Ilan and Papanicolaou [8] who describe var-ious properties of the equation in the subcritical regime, with part of their analysisrelying on very interesting numerical developments. Related references are by Ben-Artzi, Koch, and Saut [3] who gave sharp dispersive estimates for the biharmonicSchr¨odinger operator, Guo and Wang [11] who proved global well-posedness andscattering in H s for small data, Hao, Hsiao and Wang [12, 13] who discussed theCauchy problem in a high-regularity setting, and Segata [35] who proved scatteringin the case the space dimension is one. We refer also to Pausader [28, 29] where theenergy critical case for radially symmetrical initial data is discussed. The defocus-ing case like in (1.1) below is discussed in Pausader [28] for radially symmetricalinitial data. The focusing case, following the beautiful results of Kenig and Merle[18, 19], is settled in Pausader [29] still for radially symmetrical initial data.We focus in this paper on the study of the initial value problem for the cubicfourth-order defocusing equation in arbitrary space dimension R n , n ≥
1, withoutassuming radial symmetry for the intial data. The equation is written as i∂ t u + ∆ u + | u | u = 0 , (1.1)where u = I × R n → C is a complex valued function, and u | t =0 = u is in H , thespace of L functions whose first and second derivatives are in L . The equation iscritical when n = 8 because of the criticality of the Sobolev embedding H ⊂ L inthis dimension, and it enjoys rescaling invariance leaving the energy and ˙ H -normunchanged. Let S be the space of Schwartz functions. The theorem we prove in thispaper provides a complete picture of global well-posedness for (1.1). It is stated asfollows. Theorem 1.1.
Assume ≤ n ≤ . Then for any u ∈ H there exists a globalsolution u ∈ C ( R , H ) of (1.1) with initial data u (0) = u . Moreover, for any t ∈ R , the mapping u (0) u ( t ) is analytic from H into itself. On the contrary, if n ≥ then the Cauchy problem for (1.1) is ill-posed in H in the sense that for any ε > , there exist u ∈ S , t ε ∈ (0 , ε ) , and u ∈ C (cid:0) [0 , ε ] , H (cid:1) a solution of (1.1) withinitial data u such that k u k H < ε while k u ( t ε ) k H > ε − . Besides, if ≤ n ≤ ,then scattering holds true in H for (1.1) and the scattering operator is analytic. The fourth-order dispersion scaling property leads to the heuristic that smoothsolutions of the free homogeneous equation have their L ∞ norm which decays like t − n . However, the situation is not so transparent and all frequency parts of thefunction have their L ∞ -norm that decays much faster, like t − n , but at a rate whichdepends on the frequency. Uniformly, the rate of decay t − n is the best possible, butit is not optimal when the solution is localized in frequency. As one will see, thereare various differences between the dispersion behaviors of second-order Schr¨odingerequations and of (1.1).Our paper is organised as follows. We fix notations in Section 2 and recallpreliminary results from Pausader [28] in Section 3. In Section 4, we prove thatthe Cauchy problem is ill-posed when n ≥
9. In order to do so we use a low-dispersion regime argument which was essentially given in Christ, Colliander andTao [6]. We also refer to Lebeau [24, 25], Alazard and Carles [1], Carles [4] andThomann [38, 39] for other results in different settings. Starting from Section 5we focus on the energy-critical case, and so on the n = 8 part of our theorem (theequation is subcritical when n ≤ H -solutions. The first scenario is that there is a self-similar-like solution.It is not consistent with conservation of energy, conservation of local mass andcompactness up to rescaling. We exclude this scenario in Section 6. The twoother scenarii are that there is a soliton-like solution or that there is a low-to-highcascade-like solution. In these two scenarii the solution is away from the L -likeregion, namely we have that h ≤ H -norm and thus createsa 7 / H -norm control we have. In Section 9, weexclude soliton-like solution by proving that it is not consistent with the frequency-localized interaction Morawetz estimates and compactness up to rescaling. The lastscenario is excluded in Section 10 by proving that any low-to-high-like solution hasan unexpected L -regularity. Then, conservation of L -norm, frequency-localizedinteraction Morawetz estimates and conservation of energy allows us to exclude thisexistence of low-to-high cascade-like cascade solutions. Finally, in Section 11, weprove the scattering part of Theorem 1.1.As a remark, with the arguments we develop here and adaptations of the analysisin Visan [40], global well-posedness and scattering in Theorem 1.1 continue to hold UBIC FOURTH-ORDER NLS 3 true when n ≥ n -dimensional energy-critical nonlinearity with total power ( n + 4) / ( n − n ≥ n ≥
5. 2.
Notations
We fix notations we use throughout the paper. In what follows, we write A . B to signify that there exists a constant C depending only on n such that A ≤ CB .When the constant C depends on other parameters, we indicate this by a subscript,for exemple, A . u B means that the constant may depend on u . Similar notationshold for & . Similarly we write A ≃ B when A . B . A .We let L q = L q ( R n ) be the usual Lebesgue spaces, and L r ( I, L q ) be the space ofmeasurable functions from an interval I ⊂ R to L q whose L r ( I, L q ) norm is finite,where k u k L r ( I,L q ) = (cid:18)Z I k u ( t ) k rL q dt (cid:19) r . When there is no risk of confusion we may write L q L r instead of L q ( I, L r ). Twoimportant conserved quantities of equation (1.1) are the mass and the energy. Themass is defined by M ( u ) = Z R n | u ( x ) | dx (2.1)and the energy is defined by E ( u ) = Z R n (cid:18) | ∆ u ( x ) | | u ( x ) | (cid:19) dx. (2.2)In what follows we let F f = ˆ f be the Fourier transform of f given byˆ f ( ξ ) = 1(2 π ) n Z R n f ( y ) e i h y,ξ i dy for all ξ ∈ R n . The biharmonic Schr¨odinger semigroup is defined for any tempereddistribution g by e it ∆ g = F − e it | ξ | F g. (2.3)Let ψ ∈ C ∞ c ( R n ) be supported in the ball B (0 , ψ = 1 in B (0 , N = 2 k , k ∈ Z , we define the following Littlewood-Paleyoperators: \ P ≤ N f ( ξ ) = ψ ( ξ/N ) ˆ f ( ξ ) , \ P >N f ( ξ ) = (1 − ψ ( ξ/N )) ˆ f ( ξ ) , d P N f ( ξ ) = ( ψ ( ξ/N ) − ψ (2 ξ/N )) ˆ f ( ξ ) . (2.4)Similarly we define P
0, and all 1 ≤ p ≤ ∞ , independently of f , N , and p , where |∇| s is theclassical fractional differentiation operator. We refer to Tao [36] for more details.Given a ≥
1, we let a ′ be the conjugate of a , so that a + a ′ = 1.Several norms have to be considered in the analysis of the critical case of (1.1).For I ⊂ R an interval, they are defined as k u k M ( I ) = k ∆ u k L n +4) n − ( I,L n ( n +4) n ) , k u k W ( I ) = k∇ u k L n +4) n − ( I,L n ( n +4) n − n +8 ) , k u k Z ( I ) = k u k L n +4) n − ( I,L n +4) n − ) , and k u k N ( I ) = k∇ u k L ( I,L nn +2 ) . (2.6)Accordingly, we let M ( R ) be the completion of S ( R n +1 ) with the norm k · k M ( R ) ,and M ( I ) be the set consisting of the restrictions to I of functions in M ( R ). Weadopt similar definitions for W , Z , and N . We also need the following strongernorms in order to fully exploit the Strichartz estimates in Section 3. Followingstandard notations, we say that a pair ( q, r ) is Schr¨odinger-admissible, for shortS-admissible, if 2 ≤ q, r ≤ ∞ , ( q, r, n ) = (2 , ∞ , q + nr = n . (2.7)We define the full Strichartz norm of regularity s by k u k ˙ S s ( I ) = sup ( a,b ) X N N s + a k P N u k L a ( I,L b ) ! , (2.8)where the supremum is taken over all S -admissible pairs ( a, b ) as in (2.7), s ∈ R and I ⊂ R is an interval. We also define the dual norm, k h k ˙¯ S s ( I ) = inf ( a,b ) X N N s − a k P N h k L a ′ ( I,L b ′ ) ! (2.9)where again, the infimum is taken over all S -admissible pairs ( a, b ) as in (2.7), s ∈ R , and I is an interval. We let ˙ S s ( I ) be the set of tempered distributions offinite ˙ S s ( I )-norm. Finally, for a product π = Π i a i , we use the notation O ( π ) todenote an expression which is schematically like π , i.e. that is a finite combinationof products π ′ = Π i b i where in each π ′ , each b i stands for a i or for ¯ a i .As a remark, if n = 8, then there is a rescaling invariance rule for (1.1) given by u τ ( h,t ,x ) u = h u ( h ( t − t ) , h ( x − x )) (2.10) UBIC FOURTH-ORDER NLS 5 which sends a solution of (1.1) with initial data u (0) = u to another solution withdata at time t = t given by g ( h,x ) u = h u ( h ( x − x )) , (2.11)and which leaves the energy and ˙ H -norm unchanged: E (cid:0) τ ( h,t ,x ) u (cid:1) = E ( u ) and (cid:13)(cid:13) g ( h,x ) u (cid:13)(cid:13) ˙ H = k u k ˙ H for all u , u, h, t , x . The associated loss of compactness makes that (1.1) is partic-ularly difficult to handle in the critical dimension n = 8. In the radially symmetricalcase the difficulty was overcome in Pausader [28]. We prove here that we can getrid of the radially symmetrical assumption.3. Preliminary results
We recall results from Pausader [28]. We refer to Pausader [28] for their proof.A first result from Pausader [28] is that the following fundamental Strichartz-typeestimates hold true. Note that these estimates, because of the gain of derivatives,contradict the Galilean invariance one could have expected for the fourth orderSchr¨odinger equation.
Proposition 3.1.
Let u ∈ C ( I, H − ) be a solution of i∂ t u + ∆ u + h = 0 , (3.1) and u (0) = u . Then, for any S -admissible pairs ( q, r ) and ( a, b ) as in (2.7) , andany s ∈ R , k|∇| s u k L q ( I,L r ) . (cid:16) k|∇| s − q u k L + k|∇| s − q − a h k L a ′ ( I,L b ′ ) (cid:17) (3.2) whenever the right hand side in (3.2) is finite. A consequence of the Strichartz estimates (3.2) and of the commutation prop-erties of the linear propagator e it ∆ is the following estimate, for any solution u asabove: k u k ˙ S s ( I ) . k u k ˙ H s + k h k ˙¯ S s ( I ) . k u k ˙ H s + k|∇| s − a h k L a ′ ( I,L b ′ ) , (3.3)where ( a, b ) is an S -admissible pair as in (2.7), and the norms are defined in (2.8)and (2.9) above. A preliminary version of (3.2) was obtained in Kenig, Ponce andVega [20]. Let u ∈ C ( I, ˙ H ) be defined on some interval I such that 0 ∈ I andsuch that u ∈ L loc ( I × R n ). We say that u is a solution of (1.1) provided that thefollowing equality holds in the sense of tempered distributions for all times: u ( t ) = e it ∆ u + i Z t e i ( t − s )∆ (cid:0) | u | u (cid:1) ( s ) ds. (3.4)Note that, by Strichartz estimates, if u ∈ L and | u | u ∈ L loc ( I, L ), then (3.4) isequivalent to the fact that u solves (1.1) in H − with u (0) = u .The following Propositions 3.2 and 3.3, still from Pausader [28], are importantfor the energy-critical case n = 8. Proposition 3.2 settles the question of localwell-posedness. Proposition 3.3 settles the question of stability. BENOIT PAUSADER
Proposition 3.2.
Let n = 8 . There exists δ > such that for any initial data u ∈ ˙ H , and any interval I = [0 , T ] , if k e it ∆ u k W ( I ) < δ, (3.5) then there exists a unique solution u ∈ C ( I, ˙ H ) of (1.1) with initial data u . Thissolution has conserved energy, and satisfies u ∈ ˙ S ( I ) . Moreover, k u k ˙ S ( I ) . k u k ˙ H + δ , (3.6) and if u ∈ H , then u ∈ ˙ S ( I ) ∩ ˙ S ( I ) , k u k ˙ S ( I ) . k u k L , and u has conserved mass. Besides, in this case, the solution depends continuouslyon the initial data in the sense that there exists δ , depending on δ , such that, forany δ ∈ (0 , δ ) , if k v − u k H ≤ δ , and if we let v be the local solution of (1.1) with initial data v , then v is defined on I and k u − v k ˙ S ( I ) . δ . In addition to Proposition 3.2 we also have Proposition 3.3.
Proposition 3.3.
Let n = 8 , I ⊂ R be a compact time interval such that ∈ I ,and ˜ u be an approximate solution of (1.1) in the sense that i∂ t ˜ u + ∆ ˜ u + | ˜ u | ˜ u = e (3.7) for some e ∈ N ( I ) . Assume that k ˜ u k Z ( I ) < + ∞ and k ˜ u k L ∞ ( I, ˙ H ) < + ∞ . Thereexists δ > , δ = δ (Λ , k ˜ u k Z ( I ) , k ˜ u k L ∞ ( I, ˙ H ) ) , such that if k e k N ( I ) ≤ δ , and u ∈ ˙ H satisfies k ˜ u (0) − u k ˙ H ≤ Λ and k e it ∆ (˜ u (0) − u ) k W ( I ) ≤ δ (3.8) for some δ ∈ (0 , δ ] , then there exists u ∈ C ( I, ˙ H ) a solution of (1.1) such that u (0) = u . Moreover, u satisfies k u − ˜ u k W ( I ) ≤ Cδ , k u − ˜ u k ˙ S ≤ C (Λ + δ ) , and k u k ˙ S ≤ C, (3.9) where C = C (Λ , k ˜ u k Z ( I ) , k ˜ u k L ∞ ( I, ˙ H ) ) is a nondecreasing function of its arguments. In our analysis, we need to consider ˙ H -solutions. These solutions do not sat-isfy conservation of mass. However the next proposition shows that there is stillsomething remaining from that conservation law for these solutions. Proposition3.4 shows that the local mass of a solution of (1.1) varies slowly in time providedthat the radius R is sufficiently large. We define the local mass M ( u, B ( x , R ))over the ball B ( x , R ) of a function u ∈ L loc by M ( u, B ( x , R )) = Z R n | u ( x ) | ψ (( x − x ) /R ) dx, (3.10)where, ψ is as in (2.4). Proposition 3.4 from Pausader [28], states as follows. Proposition 3.4.
Let n ≥ , and u ∈ C ( I, ˙ H ) be a solution of (1.1) . Then wehave that | ∂ t M ( u ( t ) , B ( x , R )) | . E ( u ) R M ( u ( t ) , B ( x , R )) (3.11) for all t ∈ I . UBIC FOURTH-ORDER NLS 7
We refer to Pausader [28] for a proof of the above propositions.4.
Ill-posedness results
In this section we use a quantitative analysis of the small dispersion regime toprove ill-posedness results for the cubic equation when n >
8. The idea is thatnow the equation is supercritical with repect to the regularity-setting in whichwe work, namely H . Hence one can always use rescaling arguments to makeany “separation-mechanism” between two different solutions happen sooner andsooner while making the H -norm smaller and smaller. It remains then to find twosolutions whose distance goes to ∞ as time evolves. To achieve this, we follow theproof in Christ, Colliander and Tao [6] by considering the small dispersion regime.See also Lebeau [24, 25] for previous results, and Alazard and Carles [1], Carles [4]and Thomann [38, 39] for instability results in different contexts.Before we prove our theorem, we need the following lemma concerning the smalldispersion regime. Lemma 4.1.
Let k > n/ . Then, for any φ ∈ S , there exists c > such that forany ν ∈ (0 , , there exists a unique solution w ν ∈ C ([ − T, T ] , H k ) of the problem i∂ t w + ν ∆ w + | w | w = 0 (4.1) with initial data w ν (0) = φ , where T = c | log ν | c . Besides, the solution satisfies w ν ∈ C ([ − T, T ] , H p ) for any p , and k w ν − w k L ∞ ([ − T,T ] ,H k ) . φ,k ν , (4.2) where w ( t, x ) = φ ( x ) exp (cid:0) i | φ ( x ) | t (cid:1) (4.3) is a solution of the ODE formally obtained by setting ν = 0 in (4.1) .Proof. Letting u = w ν − w , we see that u solves the Cauchy problem i∂ t u + ν ∆ u = ν ∆ w + | w | w − | w + u | ( w + u ) (4.4)with u (0) = 0. Let k > n/ w ∈ C ∞ ( S ), standard developmentsensure that there exists a unique solution u ∈ C ([ − t, t ] , H k ) to (4.4), and that u can be continued as long as k u k H k remains bounded. Besides, u ∈ C ([ − t, t ] , H p )for any p ≥ t does not depend on p ). Consequently, it sufficesto prove that there exists c > s < c | log ν | c , we have that k u ( s ) k H k ≤ ν . Now, taking derivatives ∂ α of equation (4.4), multiplying by ∂ α ¯ u ,taking the imaginary part and integrating, for all α such that | α | ≤ k , we get that ∂ s k u ( s ) k H k . k u k H k (cid:0) ν k ∆ w ( s ) k H k + k| w + u | ( w + u ) − | w | w k H k (cid:1) . (4.5)By (4.3) we see that, for p ≥ k w k H p . φ,p t p . (4.6)Independently, since H k is an algebra, we get that k| w + u | ( w + u ) − | w | w k H k . X j =0 kO (cid:16)(cid:0) w (cid:1) j u − j (cid:17) k H k . k u k H k (cid:0) k w k H k + k u k H k (cid:1) . (4.7) BENOIT PAUSADER
Now, using (4.5)–(4.7), we see that, in the sense of distributions, ∂ s k u ( s ) k H k . φ,k ν (cid:0) | s | k +4 (cid:1) + k u ( s ) k H k (cid:0) | s | k + k u ( s ) k H k (cid:1) . (4.8)An application of Gromwall’s lemma gives the bound k u ( s ) k . k,φ ν exp (cid:0) C (cid:0) | s | C (cid:1)(cid:1) (4.9)for all s such that k u ( s ) k H k ≤
1. By (4.9) we see that k u ( s ) k H k ≤ | s | ≤ c | log ν | c , c > (cid:3) Now, we are in position to prove the main theorem of this section which statesthat the flow map u u ( t ), from H into H which maps the initial data to theassociated solution fails to be continuous at 0. As a remark, note that (4.10) isfalse when n ≤ H -norm controls the energy. Theorem 4.1.
Let n > . Given ε > , there exists a solution u ∈ C ([0 , ε ] , H ) such that k u (0) k H < ε and k u ( t ε ) k H > ε − , (4.10) for some t ε ∈ (0 , ε ) . Besides, we can choose u such that u (0) ∈ S and u ∈ C ([0 , ε ] , H k ) for any k > .Proof of Theorem 4.1. For φ ∈ S and ν ∈ (0 , w ν be the solution ofequation (4.1) with initial data w ν (0) = φ . By Lemma 4.1, we see that for | s | ≤ c | log ν | c , (4.2) holds true for w as in (4.3). Now, for λ ∈ (0 , ∞ ), we let u ( ν,λ ) ( t, x ) = λ w ν ( λ t, λνx ) . (4.11)Then u ( ν,λ ) solves (1.1) with initial data u ( ν,λ ) (0 , x ) = λ φ ( λνx ). A simple calcu-lation gives k u ( ν,λ ) (0) k H = λ (2 π ) n ( λν ) − n Z R n | ˆ φ ( ξ/ ( λν )) | (1 + | ξ | ) dξ . λ ( λν ) − n (cid:16) Z R n | ˆ φ ( η ) | | λνη | dη + Z R n | ˆ φ ( η ) | dη (cid:17) . φ λ ( λν ) − n , (4.12)provided that λν ≥
1. Now, given ε >
0, and ν >
0, we fix λ = λ ν,ε = (cid:0) ε ν n − (cid:1) − n − (4.13)such that λ ( λν ) − n = ε , and λν = (cid:0) εν (cid:1) − n − >
1. Independently, by (4.3), wesee that k w ( t ) k ˙ H & φ t + O ( t ) , and, consequently, using (4.2), we get that for | s | ≤ c | log ν | c sufficiently largeindependently of ν , there holds that k w ν ( s ) k ˙ H & φ s . (4.14)Consequently, using (4.11), (4.13) and (4.14) we get that k u ( ν,λ ) ( λ − t ) k H ≥ k u ( ν,λ ) ( λ − t ) k H ≥ λ ( λν ) − n k w ν ( t ) k H & φ ε t (4.15) UBIC FOURTH-ORDER NLS 9 for t sufficiently large. Now, given ε , we let ν > ε t ν > ε − , for t ν = c | log ν | c , and ε − nn − ν n − n − < ε. (4.16)We choose λ = λ ν,ε as in (4.13). Using (4.16), we get that t ε = λ − t ν < ε , andthen (4.12) and (4.15) give (4.10). This finishes the proof. (cid:3) Reduction to three scenarii ¿From now on we start with the analysis of the energy-critical case n = 8. Inthis section we prove that the analysis can be reduced to the study of some veryspecial solutions. In order to do so, we borrow ideas from previous works developedin the context of Schr¨odinger and wave equations by Bahouri and Gerard [2], Kenigand Merle [18], Keraani [21], Killip, Tao and Visan [23], and Tao, Visan and Zhang[37]. We refer also to Pausader [30] for a similar result developed in the context ofthe L -critical fourth-order Schr¨odinger equation. For any E >
0, we letΛ( E ) = sup {k u k Z ( I ) : E ( u ) ≤ E } , (5.1)where the supremum is taken over all maximal-lifespan solutions u ∈ C ( I, ˙ H ) of(1.1) satisfying E ( u ) ≤ E . In light of Proposition 3.2 and of the Strichartz estimates(3.2), we know that there exists δ > E ≤ δ , Λ( E ) . δ E < + ∞ .Besides, Λ is clearly an increasing function of E . Hence, we can define E max = sup { E > E ) < ∞} . (5.2)The goal in Sections 5–10 is to prove that E max = + ∞ . Theorem 5.1 below is afirst step in this direction. Theorem 5.1.
Suppose that E max < + ∞ . There exists u ∈ C ( I, ˙ H ) a maximal-lifespan solution of energy exactly E max such that the Z ( I ′ ) -norm of u is infinitefor I ′ = ( T ∗ , and I ′ = (0 , T ∗ ) , where I = ( T ∗ , T ∗ ) . Besides, there exist twosmooth functions h : I → R ∗ + and x : I → R n such that K = { g ( h ( t ) ,x ( t )) u ( t ) : t ∈ I } (5.3) is precompact in ˙ H , where the transformation g ( t ) = g ( h ( t ) ,x ( t )) is as in (2.11) .Furthermore, one can assume that one of the following three scenarii holds true:(soliton-like solution) there holds I = R and h ( t ) = 1 for all t ; (double low-to-highcascade) there holds lim inf t → ¯ T h ( t ) = 0 for ¯ T = T ∗ , T ∗ , and h ( t ) ≤ for all t ;(self-similar solution) there holds I = (0 , + ∞ ) and h ( t ) = t for all t . As a remark, since E ( u ) = E max , the solution u in Theorem 5.1 is such that u = 0. Assuming Propositions 6.1, 9.1 and 10.1 which exclude the three scenarii inTheorem 5.1, the following corollary holds true. Corollary 5.1.
For any
E > , there exists C = C ( E ) such that, for any u ∈ ˙ H satisfying E ( u ) ≤ E , if u ∈ C ( I, ˙ H ) is the maximal solution of (1.1) with initialdata u (0) = u , then I = R and k u k ˙ S ( R ) ≤ C .Proof of Corollary 5.1. First, using [28, Proposition 2 . . ], we see that a bound onthe Z -norm of u implies a bound on the ˙ S -norm of u . Hence if Corollary 5.1is false, then E max < + ∞ . Applying Theorem 5.1, we find a maximal solutionsatisfying one of the three scenarii in Theorem 5.1. Then, using Propositions 6.1,9.1 and 10.1, we get a contradiction. Hence E max = + ∞ . (cid:3) Now we prove Theorem 5.1.
Proof of Theorem 5.1.
In several ways the proof is similar to the one developed inthe L -critical case in Pausader [30]. We prove the more general statement thatTheorem 5.1 holds true in any dimension n ≥ H -critical equation. In particular, this is the case when n = 8. Therefore, in thisproof, (1.1) always refers to the energy-critical equation in dimension n , and theenergy E and Λ must be replaced by E ( u ) = Z R n (cid:18) | ∆ u ( x ) | + n − n | u ( x ) | nn − (cid:19) dx andΛ( E ) = sup {k u k n +4) n − Z : E ( u ) ≤ E } , where the supremum is taken over all maximal solutions of the energy-critical equa-tion of energy less or equal to E . Besides, the definition of τ and g as in (2.10)and (2.11) and Propositions 3.2 and 3.3 refer to their n -dimensional energy-criticalcounterparts. A consequence of the precised Sobolev’s inequality in Gerard, Meyerand Oru [10] and of the Strichartz estimates (3.2) is that, for any u ∈ ˙ H , k e it ∆ u k Z ( R ) . k e it ∆ |∇| u k n − n − L n +4) n − L n +4) n − k e it ∆ |∇| u k n − L ∞ L nn − . k u k n − n − ˙ H k e it ∆ |∇| u k n L ∞ ˙ H k e it ∆ |∇| u k n ( n − L ∞ ˙ B , ∞ . k u k n − n − n ( n − ˙ H k u k n ( n − ˙ B , ∞ , (5.4)where for s = 1 ,
2, ˙ B s , ∞ is a standard homogeneous Besov space. Now, thanks to(5.4), we may follow the analysis in Bahouri and Gerard [2] and Keraani [21]. In thefollowing, we call scale-core a sequence ( h k , t k , x k ) such that for every k , h k > t k ∈ R and x k ∈ R n . Mimicking the proof in Keraani [21] we obtain that for ( v k ) k a bounded sequence in ˙ H , there exists a sequence ( V α ) α in ˙ H , and scale-cores( h αk , t αk , x αk ) such that for any α = β , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log h αk h βk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ( h αk ) (cid:12)(cid:12)(cid:12) t αk − t βk (cid:12)(cid:12)(cid:12) + h αk (cid:12)(cid:12)(cid:12) x αk − x βk (cid:12)(cid:12)(cid:12) → + ∞ (5.5)as k → + ∞ , with the property that, up to a subsequence, for any A ≥ v k = A X α =1 g ( h αk ,x αk ) (cid:16) e − i ( h αk ) t αk ∆ V α (cid:17) + w Ak (5.6)for all k , where w Ak ∈ ˙ H for all k and A , andlim A → + ∞ lim sup k → + ∞ k e it ∆ w Ak k Z = 0 . (5.7)Moreover, we have the following estimates: k e it ∆ v k k n +4) n − Z = + ∞ X α =1 k e it ∆ V α k n +4) n − Z + o (1) and ,E ( v k ) = A X α =1 E ( e − i ( h αk ) t k ∆ V α ) + k w Ak k H + o (1) (5.8) UBIC FOURTH-ORDER NLS 11 for all k , where o (1) → k → + ∞ . Let ( V, ( h k ) k , ( t k ) k , ( x k ) k ) be such that V ∈ ˙ H , and ( h k , t k , x k ) ∈ R + × R × R n is a scale-core such that h k t k has a limit l ∈ [ −∞ , + ∞ ] as k → + ∞ . We say that U is the nonlinear profile associated to( V, ( h k ) k , ( t k ) k , ( x k ) k ) if U is a solution of (1.1) defined on a neighborhood of − l ,and k U ( − h k t k ) − e − ih k t k ∆ V k ˙ H → k → + ∞ . Using the analysis in Pausader [28], it is easily seen that a nonlinearprofile always exists and is unique. Besides if E ( U ) = lim k E ( e − ih k t k ∆ V ) (5.9)is such that E ( U ) < E max , then the associated nonlinear profile U is globallydefined, and k U k ˙ S ( R ) . E ( U ) . Now, we enter more specifically into the proof of Theorem 5.1. A consequence ofProposition 3.3 is that there exists a sequence of nonlinear solutions u k such that E ( u k ) < E max , E ( u k ) → E max , and k u k k Z ( −∞ , , k u k k Z (0 , + ∞ ) → + ∞ . (5.10)We let (( h αk ) k , ( t αk ) k , ( x αk ) k ) = ( h α , z α ), V α , and w A be given by (5.6) applied to thesequence ( v k = u k (0)) k . Passing to subsequences, and using a diagonal extractionargument, we can assume that, for all α , ( h αk ) t αk has a limit in [ −∞ , ∞ ]. We let U α be the nonlinear profile associated to ( V α , h α , z α ). Suppose first that thereexists α such that 0 < E ( U α ) < E max . Then, applying (5.8) and (5.9), we see thatthere exists ε > β , E (cid:0) U β (cid:1) < E max − ε , and we get that all thenonlinear profiles are globally defined. Letting W Ak ( t ) = e it ∆ w Ak , we remark that p Ak = A X α =1 τ ( h αk ,z αk ) U α + W Ak satisfies (3.7) with e = e Ak = f ( A X α =1 τ ( h αk ,z αk ) U α + W Ak ) − A X α =1 f ( τ ( h αk ,z αk ) U α )and initial data p Ak (0) = u k (0) + o A (1), where f ( x ) = | x | n − x . First, we claim thatlim sup k k A X α =1 τ ( h αk ,z αk ) U α k Z . E max ,ε A . Indeed, we remark that when ( h αk , t αk , x αk ) and ( h βk , t βk , x βk )satisfy (5.5), then for any u , v with finite Z -norm, there holds that k| τ ( h βk ,t βk ,x βk ) v | n +12 n − τ ( h αk ,t αk ,x αk ) u k L ( R ,L ) → as k → + ∞ , where τ ( h k ,t k ,x k ) is as in (2.10). Now, since Λ is sublinear around 0,and bounded on [0 , E max − ε ], using (5.8) and (5.12), we get thatlim sup k k A X α =1 τ ( h αk ,z αk ) U α k Z = A X α =1 k U α k n +4) n − Z ! n − n +4) . A X α =1 Λ( E ( U α )) ! n − n +4) . E max ,ε A X α =1 E ( U α ) ! n − n +4) . E max ,ε . Using again (5.12), we get that k f ( A X α =1 τ ( h αk ,z αk ) U α ) − A X α =1 f ( τ ( h αk ,z αk ) U α ) k L ( R ,L ) = o A (1) (5.13)as k → + ∞ . On the other hand, using the blow-up criterion in Pausader [28,Proposition 2 . . ], and the bound k U α k Z ≤ Λ ( E ( U α )) ≤ Λ ( E max − ε ), we get that,for any α , k U α k M . E max ,ε . Using the Leibnitz and chain rules for fractional derivative in Kato [17] and Visan[40, Appendix A], we obtain that k f ( A X α =1 τ ( h αk ,z αk ) U α ) − A X α =1 f ( τ ( h αk ,z αk ) U α ) k L ( R , ˙ H n +8 n +4 , n ( n +4) n n +16 ) . A,E max ,ε . (5.14)Interpolating between (5.13) and (5.14), we get that k f ( A X α =1 τ ( h αk ,z αk ) U α ) − A X α =1 f ( τ ( h αk ,z αk ) U α ) k N = o A (1) . (5.15)Now, we claim that, letting s Ak = P Aα =1 τ ( h αk ,z αk ) U α , there holds thatlim sup k k s Ak k M . E max ,ε , (5.16)independently of A . Indeed, s Ak satisfies the equation i∂ t s Ak + ∆ s Ak + A X α =1 f ( τ ( h αk ,z αk ) U α ) = 0 , with initial data s Ak (0) = A X α =1 τ ( h αk ,z αk ) U α (0) = A X α =1 g ( h αk ,x αk ) e − i ( h αk ) t αk ∆ V α + o A (1) , and consequently (5.8) and (5.9) give that k s Ak (0) k H ≤ E (cid:0) s Ak (0) (cid:1) . E max o A (1) . UBIC FOURTH-ORDER NLS 13
Using the Strichartz estimates (3.2), (5.11) and (5.15), we get that k s Ak k M . k s Ak (0) k ˙ H + k A X α =1 f ( τ ( h αk ,z αk ) U α ) k N . E (cid:0) s Ak (0) (cid:1) + o A (1) + k f ( A X α =1 τ ( h αk ,z αk ) U α ) k N . E max o A (1) + k s Ak k n − Z k s Ak k W . E max o A (1) + k s Ak k n − Z k s Ak k Z k s Ak k M . E max ,ε o A (1) + k s Ak k M . E max ,ε o A (1) (5.17)and (5.17) proves (5.16). Independently, k f ( A X α =1 τ ( h αk ,z αk ) U α + W Ak ) − f ( A X α =1 τ ( h αk ,z αk ) U α ) k L ( R ,L ) . k W Ak k Z k W Ak k n − Z + k A X α =1 τ ( h αk ,z αk ) U α k n − Z ! . E max ,ε k W Ak k Z (cid:18) k W Ak k n − Z + 1 (cid:19) . E max ,ε k W Ak k Z (5.18)and again, using (5.16) and the product and Leibnitz rules for fractional derivatives,we get that k f ( A X α =1 τ ( h αk ,z αk ) U α + W Ak ) − f ( A X α =1 τ ( h αk ,z αk ) U α ) k L ( R , ˙ H n +8 n +4 , n ( n +4) n n +1 ) . E max ,ε . (5.19)Interpolating between (5.18) and (5.19), we obtain that k f ( A X α =1 τ ( h αk ,z αk ) U α + W Ak ) − f ( A X α =1 τ ( h αk ,z αk ) U α ) k N . E max ,ε k W Ak k n +8 Z (5.20)and (5.7), (5.15) and (5.20) show thatlim sup k k e Ak k N = o (1) (5.21)as A → + ∞ . Independently, k p Ak k W ≤ k A X α =1 τ ( h αk ,z αk ) U α k W + k W Ak k W . E max ,ε o A (1) . (5.22) Now using Proposition 3.3, (5.21) and (5.22), since p Ak (0) = u k (0) + o A (1), we getthat lim sup k k u k k n +4) n − Z . lim A → + ∞ lim sup k k p Ak k n +4) n − Z . X α k U α k n +4) n − Z . E max ,ε X α E ( U α ) . E max ,ε α , we have that V α = 0.Then Strichartz estimates (3.2) and (5.8) give that k e it ∆ u k (0) k W ≤ k e it ∆ u k (0) k M k e it ∆ u k (0) k Z . E max k e it ∆ u k (0) k Z → k → + ∞ , and Proposition 3.2 gives that k u k k Z →
0, which contradicts (5.10).Consequently, we know that there exists a scale core ( h k , t k , y k ), and V ∈ ˙ H suchthat u k (0) = g ( h k ,y k ) e − it k h k ∆ V + w k , where E ( w k ) →
0. Now, up to passing to a subsequence, we can assume that t k h k → l ∈ [ −∞ , + ∞ ]. If l ∈ R , then, replacing V by e − il ∆ V , we can assume that l = 0, and changing slightly w k , we can assume that for any k , t k = 0. We then getthat u k (0) = g ( h k ,y k ) V + o (1) in ˙ H , and in particular E ( V ) = E max . Otherwise,by time reversal symmetry, we can assume that l = −∞ , and then, we find that k e it ∆ u k (0) k Z ([0 , + ∞ )) ≤ k τ ( h k ,t k ,y k ) (cid:16) e it ∆ V (cid:17) k Z ([0 , + ∞ )) + k w k k Z ([0 , + ∞ )) ≤ k e it ∆ V k Z ([ − h k t k , + ∞ )) + o (1)= o (1) , and by standard developements, we get that, for k sufficiently large, k u k k Z ( R + ) remains bounded. Once again, this contradicts (5.10). Let U be the maximalnonlinear solution of (1.1) with initial data V , defined on I = ( − T ∗ , T ∗ ). Suppose,for example that T ∗ = + ∞ , and that k U k Z ( R + ) < + ∞ . Then, using Proposition3.3 on R + with v = U , and u = τ ( h − k , , − y k ) u k , we see that k u k k Z ( R + ) is boundeduniformly in k , which is a contradiction with (5.10). Consequently, we have that k U k Z (0 ,T ∗ ) = k U k Z ( − T ∗ , = + ∞ and E ( U ) = E max . Now, we prove the compactness property of U . In the sequel, welet N min > k u k ˙ H ≤ N min implies E ( u ) < E max / ε >
0, thereexist t , . . . , t j , j = j ( ε ), such that for any time t ∈ ( − T ∗ , T ∗ ), there exist i = i ( t ),and g ( t ) = g ( h ( t ) ,y ( t )) with the property that k u ( t i ) − g ( t ) u ( t ) k ˙ H ≤ ε . Let us applythis with ε = N min . We get a function g ( t ) = g ( h ( t ) ,y ( t )) , and a finite set of times t , . . . , t j such that for any t , there exists i satisfying k u ( t i ) − g ( t ) u ( t ) k ˙ H ≤ N min . We claim that K = { g ( t ) u ( t ) : t ∈ ( − T ∗ , T ∗ ) } is precompact in ˙ H . Suppose bycontradiction that this is not true. Then, there exist ε >
0, and a sequence s k suchthat for any k and p , k g ( s k ) u ( s k ) − g ( s p ) u ( s p ) k ˙ H > ε. (5.23) UBIC FOURTH-ORDER NLS 15
According to what we said above, and passing to a subsequence, we can assumethat there exist two times ¯ t, ¯ t ′ , and a sequence g ′ k = g ( h ′ k ,y ′ k ) such that, for any k , k u (¯ t ) − g ( s k ) u ( s k ) k ˙ H < N min , and k u (¯ t ′ ) − g ′ k u ( s k ) k ˙ H < ε . (5.24)Passing to a subsequence, it is easily seen that that ( h ′ k ) − h ( s k ) remains in acompact subset of (0 , ∞ ) and that and y ( s k ) − h ( s k ) − h ′ k y ′ k remains in a com-pact subset of R n . Hence, up to considering a subsequence, we can find g ∞ suchthat g ( s k ) ( g ′ k ) − → g ∞ strongly. Now, using (5.24) and the fact that g ( h,y ) is anisometry on ˙ H for all ( h, y ), we get that k g ( s k ) u ( s k ) − g ( s k +1 ) u ( s k +1 ) k ˙ H ≤ k g ( s k ) u ( s k ) − g ∞ u (¯ t ′ ) k ˙ H + k g ∞ u (¯ t ′ ) − g ( s k +1 ) u ( s k +1 ) k ˙ H ≤ k g ′ k u ( s k ) − g ′ k g ( s k ) − g ∞ u (¯ t ′ ) k ˙ H + k g ′ k +1 u ( s k +1 ) − g ′ k +1 g ( s k +1 ) − g ∞ u (¯ t ′ ) k ˙ H ≤ ε o (1) . Clearly, this contradicts (5.23) and proves the compactness property of K . Theremaining part follows the line of the work in Tao, Visan and Zhang [37] andKillip, Tao and Visan [23]. However, in order to obtain a low-to-high cascade(instead of a high-to-low cascade), we make the following slight modification. Weuse the notations in Killip, Tao and Visan [23], except for h ( t ) = N ( t ) − . In case Osc ( κ ) is unbounded, instead of a , we introduce the quantity b ( t ) = inf (cid:18) h ( t )inf t ≥ t h ( t ) , h ( t )inf t ≤ t h ( t ) (cid:19) . Then, if sup t ∈ J b ( t ) = + ∞ , we can find intervals on which the solution presentsarbitrarily large relative peak. In particular it becomes possible to find a solutionsatisfying the low-to-high cascade scenario. Finally, in case sup t ∈ J b ( t ) < + ∞ ,the solution has arbitrarily large oscillation, but no relative peak. Mimicking theproof in Killip, Tao and Visan [23], but changing future (resp past)-focusing timeinto future (resp past)-defocusing time, one can find a solution behaving as in theself-similar case scenario. Theorem 5.1 follows. (cid:3) The self-similar case
In this section, we deal with the easiest case in Theorem 5.1, namely, the self-similar-like solution. We prove that it is not consistent with conservation of theenergy, compactness up to rescaling, and almost conservation of the local L -normas expressed in (3.11). More precisely, we prove that the following proposition holdstrue. Proposition 6.1.
Let u ∈ C ( I, ˙ H ) be a maximal-lifespan solution such that K = { g ( t ) u ( t ) : t ∈ I } is precompact in ˙ H for some function g as in (2.11) . If n = 8 ,and I = R , then u = 0 . In particular, the self-similar scenario in Theorem 5.1 doesnot hold true.Proof. Let u ∈ C ( I, ˙ H ) be a solution as above, with I = R , and let v ( t ) = g ( t ) u ( t ).Without loss of generality, we can assume that inf I = 0 and that (0 , ⊂ I . Fix < t <
1. First, using H¨older’s inequality, we get that, for any δ > Z B ( − h ( t ) x ( t ) ,δ ) | u ( t, x ) | dx . E max δ . (6.1)Independently, let x ∈ R n , R > δ > D = B ( x , R ) \ B ( − h ( t ) x ( t ) , δ ), and D ′ = B ( x ( t )+ x /h ( t ) , R/h ( t )) \ B (0 , δ/h ( t )). Using H¨older’s inequality once again,we get that Z D | u ( t, x ) | dx = h ( t ) Z D ′ | v ( t, x ) | dx ≤ h ( t ) Z | x |≥ δh ( t ) | v ( t, x ) | dx ! Z B ( x h ( t ) + x ( t ) , Rh ( t ) ) dx ! . ǫ ( δ/h ( t )) R , (6.2)where ǫ is given by ǫ ( R ) = sup t ∈ I Z | x |≥ R | v ( t, x ) | dx. A consequence of the compactness of K as in Theorem 5.1 is that ǫ ( R ) → , as R → + ∞ . (6.3)Combining (6.1) and (6.2), we get that for any ball B R of radius R > δ , Z B R | u ( t, x ) | dx . E max δ + R ǫ ( δ/h ( t )) . (6.4)Using almost conservation of local mass, as expressed in (3.11), and (6.4), we get,for any x ∈ R and any R >
4, that the following bound at time 1 holds true M ( u (1) , B ( x , R )) . E max R + M ( u ( t ) , B ( x , R )) . E max R + (cid:16) δ + R ǫ ( δ/h ( t )) (cid:17) , (6.5)where the local mass is as in (3.10). Letting t → δ →
0, we get with (6.5) that M ( u (1) , B ( x , R )) . E max R − . (6.6)Letting R → ∞ in (6.6), we obtain k u (1) k L = 0 . (6.7)Clearly (6.7) contradicts u = 0. This proves Proposition 6.1. (cid:3) An interaction Morawetz estimate
To deal with the remaining two scenarii in Theorem 5.1, in which there is noprescribed finite-time blow-up, we need a new ingredient that bounds the amountof nonlinear presence of the solution at a given scale. Natural candidates to achievethis are Morawetz estimates and in our case, interaction Morawetz estimates. Inlight of Theorem 5.1, we need to work exclusively with ˙ H -solutions. InteractionMorawetz estimates scale like the ˙ H -norm. Because of this 7 / UBIC FOURTH-ORDER NLS 17 priori interaction estimate that applies to all solutions u ∈ C ( H ), and in Section 8we use it to obtain a frequency-localized version of these estimates. The frequencylocalized version applies only to the special ˙ H -solutions given by Theorem 5.1. Weprove here that the following proposition holds true. Proposition 7.1.
Let n ≥ and let u ∈ C ([ T , T ] , H ) be a solution of (3.1) ,with forcing term h ∈ ˙¯ S ([ T , T ]) + ˙¯ S ([ T , T ]) . Then the following estimate holdstrue: n X j =1 Z T T Z R n { h, u } m ( t, y ) ( x − y ) j | x − y | { ∂ j u, u } m ( t, x ) dxdydt + n X j =1 Z T T Z R n | u ( t, y ) | ( x − y ) j | x − y | { h, u } jp ( t, x ) dxdydt + Z T T Z R n | u ( t, x ) | | u ( t, y ) | | x − y | dxdydt . sup t = T ,T k u ( t ) k L k u ( t ) k H , (7.1) where { , } m and { , } p are the mass and momentum brackets. In this proposition, the mass and momentum brackets are defined by { f, g } m = Im( f ¯ g ) , and , { f, g } p = Re( f ∇ ¯ g − g ∇ ¯ f ) . (7.2)In addition to Proposition 7.1, in order to exploit the bound given in (7.1), we alsoprove that the following lemma holds true. Lemma 7.1.
Assume n ≥ . Then k|∇| − n − u k L ≃ k ( X N N − n − | P N u | ) k L . k|∇| − n − | u | k L , (7.3) for all u ∈ ˙ H such that |∇| − | u | ∈ L , where the summation is over all dyadicnumbers.Proof. The equivalence of norms is classical. We first claim that for any g ∈ S , andany n ≥ k|∇| − n − g k L . k|∇| − n − | g | k L . (7.4)We prove (7.4). Let φ ( ξ ) = | ξ | − n − ( ψ ( ξ ) − ψ (2 ξ )) where ψ is as in (2.4). Usingthe Cauchy-Schwartz inequality we get that for any dyadic N , (cid:16) P N |∇| − n − g (cid:17) ( x )= N − n − (cid:0) g ∗ F − ( φ ( ξ/N )) (cid:1) ( x )= N n +54 Z R n g ( x − y ) ˇ φ ( N y ) dy ≤ N n +54 (cid:18)Z R n | g ( x − y ) | | ˇ φ ( N y ) | dy (cid:19) (cid:18)Z R n | ˇ φ ( N y ) | dy (cid:19) . N n +54 (cid:18)Z R n | g ( x − y ) | | ˇ φ ( N y ) | dy (cid:19) (7.5) uniformly in N . Since φ ∈ S , for any y ∈ R n , we get X N ( N | y | ) n +52 | ˇ φ ( N y ) | . X N ( N | y | ) n +52 (1 + N | y | ) − n . , (7.6)where the summation is over all dyadic numbers N . Consequently, using (7.5),(7.6) and the fact that ˇ φ ∈ S , we get that X N | P N |∇| − n − g | ( x ) . X N N n +52 Z R n | g ( x − y ) | | ˇ φ ( N y ) | dy . Z R n | g ( x − y ) | | y | n +52 X N ( N | y | ) n +52 | ˇ φ ( N y ) | ! dy . (cid:16) |∇| − n − | g | (cid:17) ( x ) , (7.7)and using the Littlewood-Paley Theorem, (7.7) gives (7.4) for g smooth. Densityarguments then give (7.3). This ends the proof of Lemma 7.1. (cid:3) Proof of Proposition 7.1.
Since the estimate we want to prove is linear, we canassume that u is smooth and use density arguments to recover the general case. Weadopt the convention that repeated indices are summed. Given some real function a , we define the Morawetz action centered at 0 by M a ( t ) = 2 Z R n ∂ j a ( x )Im(¯ u ( t, x ) ∂ j u ( t, x )) dx. (7.8)Following the computation in Pausader [28], we get that ∂ t M a ( t ) =2 Z R n (cid:16) ∂ j u∂ k ¯ u∂ jk ∆ a − (cid:0) ∆ a (cid:1) | u | − ∂ jk a∂ ik u∂ ij ¯ u + ∆ a |∇ u | + ∂ j a { u, h } jp (cid:17) dx. (7.9)Similarly, we define the Morawetz action centered at y , M ya ( t ) = M a y ( t ) for a y ( x ) = | x − y | . Finally, we define the interaction Morawetz action by the following formula: M i ( t ) = Z R n | u ( t, y ) | M ya ( t ) dy = 2Im (cid:18)Z R n Z R n | u ( t, y ) | x − y | x − y | ∇ u ( t, x )¯ u ( t, x ) dxdy (cid:19) . (7.10)We can directly estimate | M i ( t ) | ≤ k u k L ∞ L k u k L ∞ ˙ H . (7.11)Now, we get an estimate on the variation of M i by writing that ∂ t M i =2 Z R n { u, h } m ( y ) M ya dy + 4Im Z R n ∂ j u ( y ) ∂ jk ¯ u ( y ) ∂ k M ya dy + 2Im (cid:18)Z R n ¯ u ( y ) ∇ u ( y ) ∇ ∆ M ya dy (cid:19) + Z R n | u ( y ) | ∂ t M ya dy. (7.12) UBIC FOURTH-ORDER NLS 19
This gives that ∂ t M i =4 Z R n × R n Im (¯ u ( y ) ∂ j u ( y )) ∂ yj ∆ ( ∂ xk a ( x − y )) Im ( ∂ k u ( x )¯ u ( x )) dxdy + 8 Z R n × R n Im ( ∂ i u ( y ) ∂ ij ¯ u ( y )) ∂ yj ( ∂ xk a ( x − y )) Im ( ∂ k u ( x )¯ u ( x )) dxdy + 4 Z R n × R n { u, h } m ( y ) ∂ xk a ( x − y )Im ( ∂ k u ( x )¯ u ( x )) dxdy + 4 Z R n × R n | u ( y ) | ∂ xjk (∆ a ( x − y )) ∂ j u ( x ) ∂ k ¯ u ( x ) dxdy − Z R n × R n | u ( y ) | (cid:0) ∆ a ( x − y ) (cid:1) | u ( x ) | dxdy − Z R n × R n | u ( y ) | (cid:0) ∂ xjk a ( x − y ) (cid:1) ∂ ik u ( x ) ∂ ij ¯ u ( x ) dxdy + 2 Z R n × R n | u ( y ) | (cid:0) ∆ a ( x − y ) (cid:1) |∇ u ( x ) | dxdy + 2 Z R n × R n | u ( y ) | ∂ xj a ( x − y ) { u, h } jp ( x ) dxdy, (7.13)where ∂ xj denotes derivation with respect to x j , and ∂ yk derivation with respect to y k . Most of the terms in (7.13) have the right sign if we let a ( z ) = | z | . Now wefocus on the first two terms in (7.13). In the sequel, we let z = x − y . Using thefact that Re ( AB ) = Re ( A ) Re ( B ) − Im ( A ) Im ( B ), we get the equality: Z R n Im (¯ u ( y ) ∂ j u ( y )) (cid:0) ∂ yj ∂ xk ∆ a ( z ) (cid:1) Im ( ∂ k u ( x )¯ u ( x )) dxdy = − Z R n | u ( y ) | ∆ a ( z ) | u ( x ) | dxdy − R (( ∇ u ⊗ u ); ( ∇ u ⊗ u )) , (7.14)where we let R be the bilinear form on S ( R n , R n ) ⊗ S ( R n , R ) defined by R (( ~α ⊗ β ); ( ~γ ⊗ δ )) = Re Z R n α j ( y )¯ δ ( y ) (cid:0) ∂ xjk ∆ a ( z ) (cid:1) ¯ γ k ( x ) β ( x ) dxdy. (7.15)For the second term, we proceed as follows: Z R n Im ( ∂ ij ¯ u ( x ) ∂ i u ( x )) (cid:0) ∂ xj ∂ yk a ( z ) (cid:1) Im ( ∂ k u ( y )¯ u ( y )) dxdy = 14 Z R n |∇ u ( x ) | ∆ a ( z ) | u ( y ) | dxdy + Q (( ∇ ∂ i u ⊗ u ); ( ∇ u ⊗ ∂ i u )) , (7.16)where we define the quadratic form Q on S ( R n , R n ) ⊗ S ( R n , R ) by Q (( ~α ⊗ β ); ( ~γ ⊗ δ )) = Re Z R n α k ( x )¯ δ ( x ) 1 | z | (cid:18) δ jk − z j z k | z | (cid:19) ¯ γ j ( y ) β ( y ) dydx. (7.17)As one can check by computing the Fourier transform of its kernel, Q is nonnegative.Hence, applying the Cauchy-Schwartz inequality, we get | Q (( ∇ ∂ i u ⊗ u ); ( ∇ u ⊗ ∂ i u )) | ≤ | Q (( ∇ ∂ i u ⊗ u ) ) | | Q (( ∇ u ⊗ ∂ i u ) ) | ≤ Q (( ∇ ∂ i u ⊗ u ) ) + 12 Q (( ∇ u ⊗ ∂ i u ) ) (7.18) and if R and Q are as in (7.15) and (7.17), we observe that Q (( ∇ u ⊗ ∂ i u ) ) = Q (( ∇ ∂ i u ⊗ u ) ) − R (cid:0) ( ∇ u ⊗ u ) (cid:1) + 2Re Z R n ∂ k u ( x )¯ u ( x ) (cid:0) ∂ xijk a ( z ) (cid:1) ∂ ij ¯ u ( y ) u ( y ) dxdy = Q (( ∇ ∂ i u ⊗ u ) ) + R (( ∇ u ⊗ u ) )+ Re Z R n | u ( x ) | (cid:0) ∂ xij ∆ a ( z ) (cid:1) ∂ i ¯ u ( y ) ∂ j u ( y ) dxdy. (7.19)Consequently, applying (7.14), (7.16), (7.18) and (7.19), we get that4 Z R n × R n Im (¯ u ( y ) ∂ j u ( y )) ∂ yj ∆ ( ∂ xk a ( x − y )) Im ( ∂ k u ( x )¯ u ( x )) dxdy + 8 Z R n × R n Im ( ∂ i u ( y ) ∂ ij ¯ u ( y )) ∂ yj ( ∂ xk a ( x − y )) Im ( ∂ k u ( x )¯ u ( x )) dxdy ≤ − Z R n | u ( y ) | (cid:0) ∆ a ( z ) (cid:1) | u ( x ) | dxdy + 8 Q (cid:0) ( ∇ ∂ i u ⊗ u ) (cid:1) + 2 Z R n | u ( y ) | (cid:0) ∆ a ( z ) (cid:1) |∇ u ( x ) | dxdy + 4Re Z R n | u ( x ) | (cid:0) ∂ xij ∆ a ( z ) (cid:1) ∂ i ¯ u ( y ) ∂ j u ( y ) dxdy. (7.20)Now, for e ∈ R n a vector, and u a function, we define ∇ e u = ( e · ∇ u ) e | e | , and , ∇ ⊥ e u = ∇ u − ∇ e u. Then, applying the Cauchy-Schwartz inequality, we get that Q (( ∇ ∂ i u, u ) )= Z R n ∂ ij ¯ u ( x ) u ( x ) 1 | x − y | (cid:18) δ jk − ( x − y ) j ( x − y ) k | x − y | (cid:19) ∂ ik u ( y )¯ u ( y ) dxdy = Z R n (cid:2) u ( x ) ∇ ⊥ x − y ∂ i u ( y ) (cid:3) · (cid:2) ∇ ⊥ x − y ∂ i ¯ u ( x )¯ u ( y ) (cid:3) | x − y | dxdy ≤ Z R n | u ( x ) | | x − y | |∇ ⊥ x − y ∂ i u ( y ) | dxdy ≤ Z R n | u ( x ) | | x − y | (cid:18) δ jk − ( x − y ) j ( x − y ) j | x − y | (cid:19) ∂ ik ¯ u ( y ) ∂ ij u ( y ) dxdy. (7.21) UBIC FOURTH-ORDER NLS 21
Finally, (7.13), (7.20), and (7.21) give ∂ t M i ≤ − Z R n × R n | u ( y ) | (cid:0) ∆ a ( x − y ) (cid:1) | u ( x ) | dxdy + 4 Z R n × R n { u, h } m ( y ) ∂ xk a ( x − y )Im ( ∂ k u ( x )¯ u ( x )) dxdy + 2 Z R n × R n | u ( y ) | ∂ xj a ( x − y ) { u, h } jp ( x ) dxdy + 8 Z R n × R n | u ( y ) | (cid:0) ∂ xjk ∆ a ( x − y ) (cid:1) ∂ j u ( x ) ∂ k ¯ u ( x ) dxdy + 4 Z R n × R n | u ( y ) | (cid:0) ∆ a ( x − y ) (cid:1) |∇ u ( x ) | dxdy . (7.22)Let T T n −
1) ( T T − Z R n | u ( y ) | | x − y | (cid:18) ( n − δ jk − x − y ) j ( x − y ) k | x − y | (cid:19) ∂ j u ( x ) ∂ k ¯ u ( x ) dxdy which is nonpositive when n ≥
7. Finally, (7.22) and this remark give (7.1). (cid:3) A frequency-localized interaction Morawetz estimate
The preceding interaction Morawetz estimate is ill-suited for ˙ H -solutions. Inorder to exploit such an estimate in the context of ˙ H -solutions, we need to localizeit at high frequencies. The difficulty then is to deal with an inequality that scales likethe ˙ H -norm, while using only bounds that scale like the ˙ H -norm. To overcomethis difference of 7 / Proposition 8.1.
Let n = 8 . Let u ∈ C ( I, ˙ H ) be a maximal lifespan-solutionof (1.1) such that K = { g ( t ) u ( t ) : t ∈ I } is precompact in ˙ H and such that ∀ t ∈ I, h ( t ) ≤ h (0) = 1 . Then, for any sufficiently small ε > , k|∇| − | P ≥ u | k L ( I,L ) . ε , k P ≥ u k ˙ S − ( I ) . ε , and k P ≤ u k ˙ S ( I ) . ε (8.1) up to replacing u by g ( N, u for some N .Proof. We fix ε > N a dyadic number and for all time, k P ≤ N g − ( t ) ( g ( t ) u ( t )) k ˙ H = k P ≤ Nh ( t ) ( g ( t ) u ( t )) k ˙ H . (8.2)Hence, by compactness of K , and since h ≤
1, we have that k P ≤ N u k L ∞ ˙ H → N →
0. Let N be such that k P ≤ ε − N u k L ∞ ˙ H ≤ ε . Replacing K by Kg ( ε N − , , and modifying slighly h , one can assume that k P ≤ u k L ∞ ( I, ˙ H ) ≤ ε , and k P ≥ u k L ∞ ( I, ˙ H s ) ≤ k P ≤· <ε − u k L ∞ ( I, ˙ H ) + ε − s ) k P ≥ ε − u k L ∞ ( I, ˙ H ) ≤ ε, (8.3)for s ≤ /
4. We let J ( C ) = { t ≥ k|∇| − | P ≥ u | k L ([0 ,t ] ,L ) ≤ Cη } . (8.4)The first step in the proof is to obtain good Strichartz controls on the high andlow-frequency parts of u . In the sequel, we let u l = P < u , and u h = P ≥ u . Besidesthe summations are always over all dyadic numbers, unless otherwise specified. Weclaim that for J = J (2), we have that k|∇| − | P ≥ u | k L ( J,L ) ≤ η, k P ≤ u k ˙ S ( J ) . ε , and k P ≥ u k ˙ S − ( J ) . η, (8.5)provided that ε > ε < η . In the following, allspace-time norms are taken on the interval J . Applying the Strichartz estimates(3.3), we get that k P ≤ u k ˙ S . k P ≤ u (0) k ˙ H + k|∇| P ≤ (cid:0) | u l | u l (cid:1) k L ( J,L ) + X j =0 k|∇| P ≤ O (cid:16) u jl u − jh (cid:17) k L ( J,L ) . ε + k u l k S + X j =0 k|∇| P ≤ O (cid:16) u jl u − jh (cid:17) k L ( J,L ) . (8.6)Now, we estimate the terms in the sum. First, using the Bernstein’s properties(2.5) and (8.3), we get that k|∇| P ≤ O (cid:0) u l u h (cid:1) k L L . k u l u h k L L . k u l k L L k u l k L L k u h k L ∞ L . ε k u l k S . (8.7)For the next term, we remark that if N ≥ M and N ≥
8, then the Fourier supportof P N uP M v is supported in {| ξ | ≥ } , and P ≤ ( P N uP M v ) = 0. Using this remark, UBIC FOURTH-ORDER NLS 23 the Bernstein’s properties (2.5), (8.3) and the Cauchy-Schwartz inequality, we get k|∇| P ≤ O (cid:0) u l u h (cid:1) k L L . k P ≤ O (cid:0) u l u h (cid:1) k L L . X M ≤ ,N ≤ k P ≤ (cid:0) P M uP N O (cid:0) u h (cid:1)(cid:1) k L L . X M ≤ k P M u k L ∞ L X N ≤ k P N O (cid:0) u h (cid:1) k L L . X M ≤ M − k P M u k L ∞ L X N ≤ N − k P N O (cid:0) u h (cid:1) k L L . k u l k L ∞ ˙ H k|∇| − | u h | k L L . εη, (8.8)where we have used in the last inequalities that since |∇| − has a positive kernel, wehave that k|∇| − O ( u h ) k L L ≤ k|∇| − | u h | k L L . We treat the last term similarlyas follows, by writing that k|∇| P ≤ O (cid:0) u h (cid:1) k L L . k P ≤ O (cid:0) u h (cid:1) k L L . X ≤ N ≤ ,M ≤ k P ≤ (cid:0) P N u h P M O (cid:0) u h (cid:1)(cid:1) k L L + X N ≥ , N ≥ M ≥ N/ k P ≤ (cid:0) P N u h P M O (cid:0) u h (cid:1)(cid:1) k L L . X M M k P M u h k L ∞ L ! X M M − k P M | u h | k L L ! . k u h k L ∞ H k|∇| − | u h | k L L . εη. (8.9)Finally, we get with (8.6)–(8.9) that k u l k ˙ S . k P ≤ u k ˙ S . ε + ηε + ε k u l k S + k u l k S (8.10)and this proves the second inequality in (8.5) with u l instead of P ≤ u if ε > u h . Still using the Strichartz estimates (3.3) and Sobolev’sinequality, we get that k u h k ˙ S − . k u h (0) k ˙ H − + X j =0 k|∇| − P ≥ O (cid:16) u jh u − jl (cid:17) k L L . ε + X j =2 , k P ≥ O (cid:16) u jh u − jl (cid:17) k L L + X j =0 , k|∇| − P ≥ O (cid:16) u jh u − jl (cid:17) k L L . (8.11) By convolution estimate, letting c N = N − | P N u h | , we get that (cid:12)(cid:12) | u h | u h (cid:12)(cid:12) . | X M ≥ M ≥ M O ( P M u h P M u h P M u h ) | . | X M ≥ M ≥ M c M (cid:18) M M (cid:19) c M (cid:18) M M (cid:19) M P M u h | . X M c M ! (cid:18) sup M M | P M u h | (cid:19) . (8.12)Consequently, using the Bernstein’s properties (2.5), (7.3) and (8.12), we get that k P ≥ | u h | u h k L L . k| u h | u h k L L . k ( X M M − | P M u h | ) k L L (cid:18) k sup M |∇| P M u h k L ∞ L (cid:19) . k|∇| − | u h | k L L k u h k L ∞ ˙ H . E max η. (8.13)Note that instead of using the pointwise evaluation of u h = P P M u h , we canreplace u h by an arbitrary Schwartz function, get the bound, and then use densityarguments to recover (8.13). When j = 2, we proceed as follows, using Sobolev’sinequality, the Bernstein’s properties (2.5), (8.3) and the estimate for u l in (8.5), kO (cid:0) u h u l (cid:1) k L L . k u l k L L k u h k L L k u h k L ∞ L . ε k u l k ˙ S k|∇| − u h k L L k|∇| u h k L ∞ L . ε k u h k ˙ S − . (8.14)When j = 1, we proceed similarly to get k|∇| − P ≥ O (cid:0) u l u h (cid:1) k L L . kO (cid:0) u l u h (cid:1) k L L . k u l k L L k u h k L ∞ L . ε , (8.15)and finally, k|∇| − P ≥ | u l | u l k L L . k|∇|| u l | u l k L L . k u l k S . ε . (8.16)Combining (8.11) and (8.13)–(8.16), we get that k u h k ˙ S − . ε + η + ε + ε k u h k ˙ S − . η. This ends the proof of of (8.5). As a consequence of conservation of energy, (8.3),(8.5) and Hardy-Littlewood-Sobolev’s inequality, we get the following estimates on
UBIC FOURTH-ORDER NLS 25 J = J (2). Namely, k u h k L L . E max η , k u h k L L . E max η , k u h k L L . E max η , and k| u h | ∗ | x | − k L L . k u h k L L . E max ε η (8.17)Now that we have good Strichartz control on the high and low frequencies, wecan control the error terms arising in the frequency-localized interaction Morawetzestimates. First, we treat the terms arising from the mass bracket. We claim thaton J = J (2), as defined above, we have that Z J Z R n { P ≥ (cid:0) | u | u (cid:1) , u } m ( t, y ) ( x − y ) j | x − y | { ∂ j u, u } m ( t, x ) dxdy . ε η . (8.18)Exploiting cancellations, we write { P ≥ (cid:0) | u | u (cid:1) , u h } m = { P ≥ (cid:0) | u | u − | u h | u h (cid:1) , u h } m − { P < (cid:0) | u h | u h (cid:1) , u h } m + {| u h | u h , u h } m . (8.19)The last term in the right-hand side of (8.19) vanishes. Using the Bernstein’sproperties (2.5), (8.3) and (8.17), we get that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n Im ( ∂ k u h ( x )¯ u h ( x )) ( x − y ) k | x − y | { P < | u h | u h , u h } m ( y ) dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . k u h k L ∞ L k∇ u h k L ∞ L Z J (cid:12)(cid:12)(cid:0) P < | u h | u h (cid:1) u h (cid:12)(cid:12) dxdt . ε k u h k L L k P < | u h | u h k L L . ε k u h k L L k| u h | u h k L L . ε k u h k L L k u h k L ∞ L . ε η . (8.20)As for the first term in (8.19), using (8.3), we get that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n { ∂ k u h , u h } m ( x ) ( x − y ) k | x − y | { P ≥ (cid:0) | u | u − | u h | u h (cid:1) , u h } m ( y ) dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . X j =0 k u h k L ∞ H Z J (cid:12)(cid:12)(cid:12)(cid:16) P ≥ O (cid:16) u jh u − jl (cid:17)(cid:17) u h (cid:12)(cid:12)(cid:12) dxdt (8.21)and, using the Bernstein’s properties (2.5), (8.5), and (8.17), we obtain Z J (cid:12)(cid:12) P ≥ O (cid:0) u h u l (cid:1) u h (cid:12)(cid:12) dxdt . k u h k L L k u h u l k L L . k u h k L L k u l k L L ∞ . η ε. (8.22)Similarly, Z J (cid:12)(cid:12) P ≥ O (cid:0) u h u l (cid:1) u h (cid:12)(cid:12) dxdt . k u h k L L k u l k L L . η ε . (8.23)In order to treat the last term, we remark that, in view of the Fourier support, if M , M , M ≤ /
8, then P ≥ ( P M uP M uP M u ) = 0. Consequently, letting c M = M k P M u k L L and d M = M k P M u k L ∞ L , we get, using again the Bernstein’sproperties (2.5), (8.3) and (8.5), that Z J (cid:12)(cid:12)(cid:0) P ≥ | u l | u l (cid:1) u h (cid:12)(cid:12) dxdt . k u h k L ∞ L k P ≥ X ≥ M ≥ M ≥ M P M uP M uP M u k L L . k u h k L ∞ L X ≥ M ≥ / ,M ≥ M ≥ M k P M uP M uP M u k L L . k u h k L ∞ L X ≥ M ≥ / ,M ≥ M ≥ M k P M k L L k P M u k L L k P M u k L ∞ L ∞ . k u h k L ∞ L X ≥ M ≥ / k P M u k L L X ≥ M ≥ M c M d M (cid:18) M M (cid:19) . k u h k L ∞ L k u l k S . ε . (8.24)Combining (8.19)–(8.24), we see that (8.18) holds true. Now, we turn to the lasterror term, which arises from the momentum bracket. We claim that on J = J (2),we have that (cid:12)(cid:12)(cid:12) Z J Z R n | u h ( s, y ) | ( x − y ) j | x − y | { P ≥ | u | u, u h } jp ( s, x ) dxdyds − Z J Z R n | u h ( s, y ) | | u h ( s, x ) | | x − y | dxdyds (cid:12)(cid:12)(cid:12) . η (cid:16) ε η − + ε + ε η − (cid:17) . (8.25)In order to prove (8.25), we decompose { P ≥ | u | u, u h } p = {| u | u, u } p − {| u l | u l , u l } p − { (cid:0) | u | u − | u l | u l (cid:1) , u l } p − { P < | u | u, u h } p = − ∇ (cid:0) | u | − | u l | (cid:1) − { (cid:0) | u | u − | u l | u l (cid:1) , u l } p − { P < | u | u, u h } p . (8.26)Besides, we remark that { f, g } p = ∇O ( f g ) − O ( f ∇ g ) . (8.27)Now, we estimate R = X k =0 Z J Z R n | u h ( s, y ) | ( x − y ) j | x − y | {O (cid:0) u kl u − kh (cid:1) , u l } jp ( s, x ) dxdyds. (8.28) UBIC FOURTH-ORDER NLS 27
The case k = 2 is treated as follows, using (8.27). First (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | ( x − y ) j | x − y | O (cid:0) u h u l (cid:1) ∂ j u l ( x ) dsdxdy (cid:12)(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | O Z R n (cid:0) |∇| − u h (cid:1) (cid:18) |∇| (cid:18) ( x − y ) j | x − y | (cid:0) u l ∂ j u l (cid:1) ( x ) (cid:19)(cid:19) dxdsdy (cid:12)(cid:12)(cid:12)(cid:12) . (8.29)Now, using the boundedness of the Riesz transform and the Bernstein’s properties(2.5), we estimate for any y ∈ R n , k|∇| (cid:18) ( x − y ) j | x − y | (cid:0) u l ∂ j u l (cid:1) ( x ) (cid:19) k L . k∇ (cid:18) ( x − y ) j | x − y | (cid:0) u l ∂ j u l (cid:1) ( x ) (cid:19) k L . k| x − y | − {| x − y |≤ } k L k u l k L ∞ k∇ u l k L + k| x − y | − {| x − y |≥ } k L ∞ k u l k L k∇ u l k L + k∇ u l k L k ∂ j u l k L k u l k L + k u l k L k∇ u l k L . k u l k L k∇ u l k L , (8.30)where E is the characteristic function of the set E . Consequently, using the Bern-stein’s properties (2.5), (8.3), (8.5) and (8.30), we get that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | O Z R n (cid:0) |∇| − u h (cid:1) (cid:18) |∇| (cid:18) ( x − y ) j | x − y | (cid:0) u l ∂ j u l (cid:1) ( x ) (cid:19)(cid:19) dxdsdy (cid:12)(cid:12)(cid:12)(cid:12) . Z J Z R n | u h ( y ) | k|∇| − u h k L k|∇| (cid:18) ( x − y ) j | x − y | u l ∂ j u l (cid:19) k L . k u h k L ∞ L k|∇| − u h k L L k u l k L L k∇ u l k L ∞ L . ηε . (8.31)Besides, integrating by parts and using (8.5) and (8.17), we finish the analysis ofthe case k = 2 as follows: (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | − |O (cid:0) u l u h (cid:1) ( x ) | dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k| u h | ∗ | x | − k L L k u h k L L k u l k L L . k u h k L L k u h k L L k u l k S . ηε . (8.32)The case k = 1 is similar. First, with the Bernstein’s properties (2.5), (8.3), (8.5)and (8.17), we obtain that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | ( x − y ) j | x − y | O (cid:0) u h u l (cid:1) ( x ) ∂ j u l ( x ) dsdxdy (cid:12)(cid:12)(cid:12)(cid:12) . k u h k L ∞ L k u h k L L k∇ u l k L L k u l k L ∞ L . k u h k L ∞ L k u l k L ∞ L k u l k ˙ S k u h k L L . ε η (8.33) and then, (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | − O (cid:0) u h u l (cid:1) ( x ) dsdxdy (cid:12)(cid:12)(cid:12)(cid:12) . k| u h | ∗ | x | − k L L k u h k L ∞ L k u h k L ∞ L k u l k L L . k u h k L L k u h k L ∞ H k u l k L L . ε η . (8.34)Finally for the case k = 0, using the Bernstein’s properties (2.5), (8.3), (8.5) and(8.17), we write that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | ( x − y ) j | x − y | O (cid:0) u h (cid:1) ( x ) ∂ j u l ( x ) dsdxdy (cid:12)(cid:12)(cid:12)(cid:12) . k u h k L ∞ L k u h k L L k∇ u l k L L ∞ . η ε , (8.35)and that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | − O (cid:0) u h u l (cid:1) ( x ) dsdxdy (cid:12)(cid:12)(cid:12)(cid:12) k| u h | ∗ | x | − k L L k u h k L L k u h k L ∞ L k u l k L L . ε η . (8.36)This finishes the analysis of the second error term in the momentum bracket (8.26),namely R . Now we turn to the third error term arising from (8.26), i.e.˜ R = X k =0 Z J Z R n | u h ( s, y ) | ( x − y ) j | x − y | { P < O (cid:0) u kh u − kl (cid:1) , u h } jp ( s, x ) dxdyds. We treat the term k = 0 using (8.27) as follows. First, we get that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | Z R n u h ( x − y ) j | x − y | ∂ j (cid:0) P < O (cid:0) u l (cid:1)(cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | Z R n (cid:0) |∇| − u h (cid:1) (cid:18) |∇| (cid:18) ( x − y ) j | x − y | ∂ j P < O (cid:0) u l (cid:1)(cid:19)(cid:19) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z J Z R n | u h ( y ) | k|∇| − u h k L k|∇| (cid:18) ( x − y ) j | x − y | ∂ j P < O (cid:0) u l (cid:1)(cid:19) k L dsdy. (8.37)Using the boundedness of the Riesz transform, we see that k|∇| (cid:18) ( x − y ) j | x − y | ∂ j P < O (cid:0) u l (cid:1)(cid:19) k L . k∇ (cid:18) ( x − y ) j | x − y | ∂ j P < O (cid:0) u l (cid:1)(cid:19) k L . k {| x − y |≤ } | x − y | − k L k ∂ j u l k L k u l k L ∞ + k {| x − y |≥ } | x − y | − k L ∞ k ∂ j u l k L k u l k L + k∇ ∂ j u l k L k u l k L + k∇ u l k L k∇ u l k L k u l k L . k u l k L k∇ u l k L , (8.38) UBIC FOURTH-ORDER NLS 29 and, consequently, using the Bernstein’s properties (2.5), (8.3), (8.5) and (8.38)above, we obtain that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | (cid:18)Z R n u h ( x ) ( x − y ) j | x − y | ∂ j (cid:0) P < | u l | u l (cid:1) ( x ) dx (cid:19) dyds (cid:12)(cid:12)(cid:12)(cid:12) . Z J Z R n | u h ( y ) | k|∇| − u h k L k∇ u l k L k u l k L dyds . k u h k L ∞ L k|∇| − u h k L L k∇ u l k L L k u l k L ∞ L . ηε . (8.39)As for the other part, using (8.5) and (8.17), we get that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | P < O (cid:0) u l (cid:1) ( x ) u h ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k| u h | ∗ | x | − k L L k u h k L L k u l k L L k u l k L ∞ L . ε η . (8.40)Now, we treat the case k = 1 using Bernstein property (2.5), (8.3), (8.5) and (8.17)as follows. First we write that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | ( x − y ) j | x − y | u h ( x ) ∂ j (cid:0) P < O (cid:0) u l u h (cid:1)(cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k u h k L ∞ L k u h k L L k u l u h k L L . k u h k L ∞ L k u h k L L k u l k L L . ε η , (8.41)and then we write that (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | u h ( x ) P < O (cid:0) u l u h (cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k| u h | ∗ | x | − k L L k u h k L L k u l k L ∞ L . ε η . (8.42)When k = 2, we use the Bernstein’s properties (2.5), (8.3), (8.5), and (8.17) to get (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | ( x − y ) j | x − y | u h ( x ) ∂ j (cid:0) P < O (cid:0) u l u h (cid:1)(cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k u h k L ∞ L k u h k L L k ∂ j P < O (cid:0) u l u h (cid:1) k L L . k u h k L ∞ L k u h k L L k u h u l k L L . k u h k L ∞ L k u h k L L k u l k L ∞ L ∞ . ε η , (8.43) and (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | u h ( x ) P < O (cid:0) u l u h (cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k| u h | ∗ | x | − k L L k u h k L L k P < O (cid:0) u l u h (cid:1) k L L . k u h k L L k u h k L L k P < O (cid:0) u l u h (cid:1) k L L . k u h k L L k u h k L L k u h k L ∞ L k u l k L ∞ L . ε η . (8.44)Finally, the case k = 3 is treated as follows using the Bernstein’s properties (2.5),(8.3), (8.5), and (8.17) (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | ( x − y ) j | x − y | u h ( x ) ∂ j (cid:0) P < O (cid:0) u h (cid:1)(cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k u h k L ∞ L k u h k L L k∇ P < O (cid:0) u h (cid:1) k L L . k u h k L ∞ L k u h k L L k u h k L L . k u h k L ∞ L k u h k L L k u h k L L . ε η (8.45)and, similarly, (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( y ) | | x − y | u h ( x ) P < O (cid:0) u h (cid:1) ( x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) . k| u h | ∗ | x | − k L L k u h k L L k P < O (cid:0) u h (cid:1) k L L . k u h k L L k u h k L L k P < O (cid:0) u h (cid:1) k L L . k u h k L L k u h k L L k u h k L ∞ L . ε η . (8.46)This finishes the analysis of ˜ R . The first error term in (8.26) is now easy to treat.Indeed, integrating by parts, (cid:12)(cid:12)(cid:12)(cid:12)Z J Z R n | u h ( s, y ) | ( x − y ) | x − y | ∇ (cid:0) | u | − | u l | − | u h | (cid:1) ( s, x ) dxdyds (cid:12)(cid:12)(cid:12)(cid:12) ≤ X k =1 O Z J Z R n | u h ( s, y ) | | u h ( s, x ) | − k | u l ( s, x ) | k | x − y | dxdyds . (8.32) + (8.34) + (8.36) . ε η . (8.47)Finally, with (8.26)–(8.47), we obtain (8.25). As a consequence of (7.1), (8.18) and(8.25) on J = J (2), we have that k|∇| − | u h | k L L . η (cid:16) ε η − + ε + ε η − (cid:17) ≤ η , (8.48)if ε > η > ε . Letting η = ε , we obtain J (2) ⊂ J (1).Finally, J (1) is a closed, open nonempty subset of R . Hence J (1) = R , and thisfinishes the proof. (cid:3) UBIC FOURTH-ORDER NLS 31
It follows from H¨older’s inequality that in the situation of Proposition 8.1, onealso has the estimates (8.17) with η = ε .9. The Soliton case
In this section, we deal with the first scenario in Theorem 5.1, namely the soli-ton case. We prove that the soliton scenario is inconsistent with the frequency-localized Morawtez interaction estimates developed in Section 7 and compactnessup to rescaling.
Proposition 9.1.
Let u ∈ C ( R , ˙ H ) be a solution of (1.1) such that K = { u ( t ) : t ∈ R } is precompact in ˙ H up to translation. If n = 8 , then u = 0 . In particularthe soliton scenario in Theorem 5.1 does not hold true.Proof. Let u ∈ C ( R , ˙ H ) be a solution of (1.1) of energy E ( u ) > K = { g (1 ,y ( t )) u ( t ) : t ∈ R } is precompact in ˙ H . In particular we can applyProposition 8.1 with ε > k|∇| − P ≥ u k L ( R ,L ) . . (9.1)Independently, by (8.1), we know that, for all t , k P ≥ u ( t ) k H & E ( u ) E ( u ) − ε > , (9.2)if ε is sufficiently small. Then (9.2) implies that for all v in the ˙ H -closure of K , P ≥ v = 0. Since K is precompact in ˙ H and the mapping v
7→ k|∇| − P ≥ v k L iscontinuous on ˙ H , we get that there exists κ > ∀ v ∈ K, k|∇| − P ≥ v k L ≥ κ. (9.3)Now, (9.1) and (9.3) imply that κ t . k|∇| − u h k L ([0 ,t ] ,L ) . . (9.4)Letting t → + ∞ , we get a contradiction in (9.4). This finishes the proof of Propo-sition 9.1. (cid:3) The Low-to-high cascade
Now, we are ready to deal with the last scenario, and to exclude the case of alow-to-high cascade solution. In order to do so, we use the estimates coming fromthe frequency-localized interaction Morawetz estimates developed in Section 7 tocontrol the action of the high-frequency part of u . Then the low-frequency partobeys an analogue of (1.1) with initial data arbitrarily small. Hence one can makeits ˙ S -norm small, depending on the frequency, so as to prove that it is in factsmall in L . Then the solution is an H solution, and conservation of mass gives acontradiction. More precisely, we prove the following proposition. Proposition 10.1.
Let u ∈ C ( I, ˙ H ) be a maximal lifespan solution of (1.1) suchthat K = { g ( h ( t ) ,x ( t )) u ( t ) : t ∈ I } is precompact in ˙ H for some functions h, x suchthat h ( t ) ≤ h (0) = 1 , and lim inf t → sup I h ( t ) = 0 , (10.1) then if n = 8 , we have that u = 0 . In particular, the low-to-high cascade scenariodoes not hold true. Proof.
Let u be as above. Applying Proposition 6.1, we see that I = R , and since h ≤
1, given ε >
0, we can apply Proposition 8.1 to get that (8.1) holds true. Wemay also suppose that (8.5) holds true. As a first step in the proof, we claim thatif ε > M ≤ k P ≤ M u k ˙ S . M . (10.2)Fix M , a dyadic number, let m = M and let κ > K is precompact, using (8.2) we get thatthere exists t > k P ≤ u ( t ) k ˙ H ≤ κ m. (10.3)We claim that for any C >
0, if κ is sufficiently small, independently of m , then wehave that, for all dyadic numbers M ∈ [ m, k P ≤ M u k ˙ S ( J ) ≤ κC (cid:0) m + M (cid:1) (10.4)when J is small and t ∈ J . Indeed, using the Bernstein’s properties (2.5), we getthat, in J , k P ≤ M u k S . X N ≤ M N k P N u k L ∞ L + X N ≤ M N k P N u k L L . X N ≤ M N k P N u ( t ) k L + | J | X N ≤ M N k ∂ t P N k L ∞ L + | J | X N ≤ M N k P N u k L ∞ L . κ m + | J | X N ≤ M N k P N ∆ u k L ∞ L + | J | X N ≤ M N k P N (cid:0) | u | u (cid:1) k L ∞ L + | J | X N ≤ M N k P N u k L ∞ L . E ( u ) κ m + M | J | + | J | X N ≤ M N k| u | u k L ∞ L + | J | M . E ( u ) κ m + M | J | + M | J | , and if | J | . E ( u ) ,C κ , then (10.4) holds true. Now, let J ( C ) be the maximum intervalcontaining t on which (10.4) holds true for the constant C >
0. We prove that J (2) ⊂ J (1) if κ and ε are chosen sufficiently small, independently of m . Indeed,let u vlow = P ≤ m u , and u med = P m< · < u. In the following, all time integrals are taken on J = J (2). Applying Strichartzestimates (3.3), we get that k P ≤ M u k ˙ S . k P ≤ M u ( t ) k ˙ H + k∇ P ≤ M | u vlow | u vlow k L L + k∇ P ≤ M (cid:0) | u | u − | u vlow | u vlow (cid:1) k L L . κm + k P ≤ m u k S + M k ˜ P ≤ M (cid:0) | u vlow | | u med | + | u vlow | | u h | (cid:1) k L L + M k ˜ P ≤ M | u med | k L L + M k ˜ P ≤ M | u h | k L L , (10.5) UBIC FOURTH-ORDER NLS 33 where ˜ P ≤ M is the convolution operator whose kernel is˜ k ( x ) = M ˆ ψ ( M x ) , where ψ is as in (2.4). We remark that ˜ P ≤ M has nonnegative kernel and satisfiesestimates similar to those of P ≤ M . In particular, (2.5) holds true with ˜ P ≤ M inplace of P ≤ M . By assumption we have that k P ≤ m u k S ≤ (4 κ ) m . (10.6)Besides, using the Bernstein’s properties (2.5), and the assumption on J , we getthat M k ˜ P ≤ M | u vlow u med |k L L . M k u vlow k L ∞ L k u med k L L . M (4 κm ) X m 1. Now, we finish the proofof Proposition 10.1. A consequence of (10.2) is that u ∈ L ∞ L . Indeed, by theBernstein’s properties (2.5), P ≥ M u ∈ L ∞ L for any dyadic M , and using (10.2),we get that, when M ≤ k P ≤ M u k L ∞ L ≤ X N ≤ M k P N u k L ∞ L . X N ≤ M N − k P ≤ N u k ˙ S . X N ≤ M N . M. (10.12)Now, let M > K is precompact in ˙ H , we can find t such that k P M< ·≤ M − u ( t ) k L ≤ M − k P M< ·≤ M − u ( t ) k ˙ H ≤ M − k P Mh ( t ) < ·≤ M − h ( t ) ( g ( t ) u ( t )) k ˙ H ≤ M. (10.13)Using conservation of mass, the Bernstein’s properties (2.5), (10.12) and (10.13),we deduce that k u (0) k L = k u ( t ) k L ≤ k P >M − u ( t ) k L + k P M< ·≤ M − u ( t ) k L + k P ≤ M u k L ∞ L ≤ M E ( u ) + 2 M. (10.14)Since M is arbitrary, we get that u (0) = 0. This concludes the proof of Proposition10.1. (cid:3) Analiticity of the flow map and scattering In view of Theorem 4.1 and Corollary 5.1, we can finish the proof of the firstassertions in Theorem 1.1 with Proposition 11.1 below. Proposition 11.1. Let n ≤ . Then, for any t > , the mapping u u ( t ) , from H into H , is analytic.Proof. We follow arguments developed in Pausader and Strauss [31] for the fourth-order wave equation. We use the implicit function theorem. In case 1 ≤ n ≤ L ∞ -norm of u , andhence, the nonlinear term is lipschitz. In this case the problem can be solved withbasic arguments. Now we treat the case n ≥ 4. We divide [0 , t ] = ∪ kj =1 I j intosubintervals I j = [ a j , a j +1 ] such that k∇ u k L n +42 ( I,L n ( n +4)3 n +4 ) ≤ δ. (11.1) UBIC FOURTH-ORDER NLS 35 First, if I = I = [0 , a ], we consider the mapping T I : H × ˙ S ( I ) ∩ ˙ S ( I ) → H × ˙ S ( I ) ∩ ˙ S ( I )defined by T ( u , v ) = (cid:18) u , t e it ∆ u + i Z t e i ( t − s )∆ | v | v ( s ) ds (cid:19) . The map T is well defined thanks to the Strichartz estimates (3.3). It is clearlyanalytic, and u ∈ C ( I, H ) is a solution of (1.1) if and only if T ( u (0) , u ) = ( u (0) , u ).An application of Strichartz estimates gives that, if δ in (11.1) is sufficiently small,then k D T ( u (0) , u ) k ˙ S ∩ ˙ S → ˙ S ∩ ˙ S < , where D denotes derivation with respect to the second argument. Consequently, D ( I − T ) ( u (0) , u ) is invertible, and the implicit function theorem ensures that u u | I is analytic. In particular, u u ( a ), from H into H , is analytic. Byfinite induction, we get that u u ( t ) is analytic. (cid:3) Now, we turn to the proof of the scattering assertion of Theorem 1.1. Thestatement is an easy consequence of Propositions 11.2 and 11.3 below. Proposition 11.2. Let ≤ n ≤ . For any u + ∈ H , respectively u − ∈ H , thereexists a unique u ∈ C ( R , H ) , solution of (1.1) such that k u ( t ) − e it ∆ u ± k H → as t → ±∞ . Besides, we have that M ( u (0)) = M ( u ± ) , and E ( u (0)) = k u ± k H . (11.3) This defines two mappings W ± : u ± u (0) from H into H , and W + and W − are continuous in H .Proof. By time reversal symmetry, we need only to prove Proposition 11.2 for u + .Let ω ( t ) = e it ∆ u + . Then by the Strichartz estimates (3.3), ω ∈ ˙ S ( R ) ∩ ˙ S ( R )and, given δ > 0, there exists T δ such that, on I = [ T δ , + ∞ ), (11.1) holds true with ω instead of u . For u ∈ ˙ S ( I ) ∩ ˙ S ( I ), we defineΦ( u )( t ) = ω ( t ) − i Z ∞ t e i ( t − s )∆ | u ( s ) | u ( s ) ds. (11.4)For δ sufficiently small, Φ defines a contraction mapping on the set X T δ = { u ∈ ˙ S ( I ) ∩ ˙ S ( I ); k∇ u k L n +42 ( I,L n ( n +4)3 n +4 ) ≤ δ, k u k ˙ S ( I ) + k u k ˙ S ( I ) . k u + k H } , equipped with the ˙ S ( I )-norm. Thus Φ admits a unique fixed point u . We observethat u ( T δ + t ) = e it ∆ u ( T δ ) + i Z T δ + tT δ e i ( t − s )∆ | u ( s ) | u ( s ) ds in H . Consequently, u solves (1.1) on I = [ T δ , + ∞ ). Hence, using the first partof Theorem 1.1, u can be extended for all times t ∈ R . Now, (11.2) follows from(11.4) and the boundedness of u in ˙ S and ˙ S -norms. Uniqueness follows from the fact that any solution of (1.1) has a restriction in X T for some T ≥ T δ , anduniqueness of the fixed point of Φ in such spaces. The continuity statements areeasy adaptations of the proof of local well-posedness, see Pausader [28]. The firstequality in (11.3) follows from conservation of Mass and convergence in L . For thesecond, we remark that since ω ∈ ˙ S ( R ) there exists a sequence of times t k → + ∞ such that k ω ( t k ) k L → 0. Then, using conservation of energy, we compute2 E ( u (0)) = 2 E ( u ( t k ))= 2 E ( ω ( t k )) + o (1)= k ω ( t k ) k H + o (1) = k u + k H + o (1) , and letting k → + ∞ we get that the second equation in (11.3) holds true. Thisfinishes the proof of Proposition 11.2. (cid:3) Proposition 11.3. Let ≤ n ≤ . Given any solution u ∈ C ( R , H ) of (1.1) , thereexist u ± ∈ H such that (11.2) holds true. In particular W ± are homeomorphismsof H .Proof. In case 5 ≤ n ≤ 7, the equation is subcritical, and standard developmentsusing the decay properties of the linear propagator, conservation of mass and theusual Morawetz estimates, give that for any solution u ∈ C ( R , H ) of (1.1), thereexists C > k u k L ( R ,L ) ≤ C. On such an assertion we refer to Cazenave [5] or Lin and Strauss [26] for thesecond order case, and to Pausader [28] for the classical Morawetz estimates in thecase of the fourth-order Schr¨odinger equation. Consequently, applying Strichartzestimates, we get that k u k ˙ S ( R ) + k u k ˙ S ( R ) . u . (11.5)In case n = 8, as a consequence of Corollary 5.1, we get that any nonlinear solution u satisfies k u k Z ( R ) . E ( u ) . Using the work in Pausader [28, Proposition 2 . n = 8. Since e it ∆ is an isometry on H , (11.2) is equivalent to provingthat there exists u + ∈ H such that k e − it ∆ u ( t ) − u + k H → t → + ∞ . Now we prove that e − it ∆ u ( t ) satisfies a Cauchy criterion. We notethat Duhamel’s formula gives that e − it ∆ u ( t ) − e − it ∆ u ( t ) = i Z t t e − is ∆ | u ( s ) | u ( s ) ds. (11.7)By duality, (3.3) gives that for any s ∈ [0 , , and any h ∈ ˙¯ S ( R ), we have that k Z R e − it ∆ h ( t ) dt k ˙ H s . k h k ˙¯ S s ( R ) . (11.8)Now, (11.5) and (11.8) give that the right hand side in (11.7) is like o (1) in H as t , t → + ∞ . In particular, e − it ∆ u ( t ) satisfies a Cauchy criterion, and there exists u + ∈ H such that (11.6) holds true. We also get that u + = u + i Z ∞ e − is ∆ | u ( s ) | u ( s ) ds, (11.9) UBIC FOURTH-ORDER NLS 37 and u + is unique. The continuity statements are easy adaptations of the proofof local well-posedness, see Pausader [28]. Now, by uniqueness, we clearly havethat u (0) = W + ( u + ), so that W + is an homeomorphism. This ends the proof ofProposition 11.3. (cid:3) Proof of the scattering in Theorem 1.1. Applying Propositions 11.2 and 11.3, wesee that the scattering operator S = W + ◦ W − − is an homeomorphism from H into H . Using (11.4) and (11.9), and adapting slightly the proof of Proposition11.1, we easily see that S is analytic. This ends the proof of the scattering part inTheorem 1.1. (cid:3) ACKNOWLEDGEMENT: The author expresses his deep thanks to Emmanuel Hebeyfor his constant support and for stimulating discussions during the preparation ofthis work. References [1] Alazard, T., and Carles, R., Loss of regularity for supercritical nonlinear Schrdinger equations,Math. Ann. to appear.[2] Bahouri, H., and Gerard, P., High frequency approximation of solutions to critical nonlinearwave equations, Amer. J. of Math. , 121, (1999), 131–175.[3] Ben-Artzi, M., Koch, H., and Saut, J.C., Dispersion estimates for fourth order Schr¨odingerequations, C.R.A.S., 330, S´erie 1, (2000), 87–92.[4] Carles, R., Geometric optics and instability for semi-classical Schrdinger equations. Arch.Ration. Mech. Anal. 183 (2007), No 3, 525–553.[5] Cazenave, T., Semilinear Schr¨odinger equations , Courant Lecture Notes in Mathematics,10, New York University, Courant Institute of Mathematical Sciences, New York; AmericanMathematical Society, Providence, RI, (2003).[6] Christ, M., Colliander, J., and Tao, T., Ill-posedness for nonlinear Schr¨odinger and waveequations, Ann. I.H.P. to appear.[7] Colliander, J., Keel, M., Staffilani, G., Takaoka, H., and Tao, T., Global well-posedness andscattering in the energy space for the critical nonlinear Schr¨odinger equation in R . Ann. ofMath. to appear.[8] Fibich, G., Ilan, B., and Papanicolaou, G., Self-focusing with fourth order dispersion. SIAMJ. Appl. Math. 62, No 4, (2002), 1437–1462.[9] Fibich, G., Ilan, B., and Schochet, S., Critical exponent and collapse of nonlinear Schr¨odingerequations with anisotropic fourth-order dispersion. Nonlinearity 16 (2003), 1809–1821.[10] Gerard, P., Meyer, Y., and Oru, F., In´egalit´es de Sobolev pr´ecis´ees. S´eminaire EDP ´Ecolepolytechnique 1996-1997, 11pp.[11] Guo, B., and Wang, B., The global Cauchy problem and scattering of solutions for nonlinearSchr¨odinger equations in H s , Diff. Int. Equ. 15 No 9 (2002), 1073–1083.[12] Hao, C., Hsiao, L., and Wang, B., Well-posedness for the fourth-order Schr¨odinger equations, J. of Math. Anal. and Appl. 320 (2006), 246–265.[13] Hao, C., Hsiao, L., and Wang, B., Well-posedness of the Cauchy problem for the fourth-orderSchr¨odinger equations in high dimensions, J. of Math. Anal. and Appl. 328 (2007) 58–83.[14] Huo, Z., and Jia, Y., The Cauchy problem for the fourth-order nonlinear Schr¨odinger equationrelated to the vortex filament, J. Diff. Equ. 214 (2005), 1–35.[15] Karpman, V. I., Stabilization of soliton instabilities by higher-order dispersion: fourth ordernonlinear Schr¨odinger-type equations. Phys. Rev. E 53, 2 (1996), 1336–1339.[16] Karpman, V. I., and Shagalov, A.G., Stability of soliton described by nonlinear Schr¨odinger-type equations with higher-order dispersion, Phys. D. 144 (2000) 194–210.[17] Kato, T., On nonlinear Schr¨odinger equations, II. H s -solutions and unconditional well-posedness. J. Anal. Math. 67 (1995), 281–306.[18] Kenig, C., and Merle, F., Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schr¨odinger equation in the radial case, Invent. Math. 166 No 3(2006), 645–675. [19] Kenig, C., and Merle, F., Global well-posedness, scattering and blow-up for the energy-criticalfocusing nonlinear wave equation. Acta Math. to appear.[20] Kenig, C., Ponce, and Vega, L., Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40, (1991), 33-69.[21] Keraani, S., On the defect of compactness for the Strichartz estimates of the Schr¨odingerequations, J. Diff. Eq. preprint .[23] Killip, R., Tao, T., and Visan, M., The cubic nonlinear Schr¨odinger equation in two dimen-sions with radial data preprint .[24] Lebeau, G., Non linear optic and supercritical wave equation, Bull. Soc. Roy. Sci. Lige Bull. Soc. Math.France. 133 (2005) 1 145–157.[26] Lin, J. E., and Strauss, W.A., Decay and scattering of solutions of a nonlinear Schr¨odingerequation. J. Funct. Anal. 30, (1978), 245–263.[27] Miao, C., Xu, G., and Zhao L., Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions d ≥ preprint [28] Pausader, B., Global well-posedness for energy critical fourth-order Schr¨odinger equations inthe radial case, Dynamics of PDE , 4 (3), (2007), 197–225.[29] Pausader, B., The focusing energy-critical fourth-order Schr¨odinger equation with radial data, preprint .[30] Pausader, B., Minimal mass blow-up solutions for the mass-critical fourth-order Schr¨odingerequation, preprint .[31] Pausader, B., and Strauss, W. A., Analyticity of the Scattering Operator for Fourth-orderNonlinear Waves, preprint .[32] Ryckman, E., and Visan, M., Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrodinger equation in R , Amer. J. Math. 129 (2007), 1–60.[33] Segata, J., Well-posedness for the fourth-order nonlinear Schr¨odinger type equation relatedto the vortex filament, Diff. Int. Equ. 16 No 7 (2003), 841–864.[34] Segata, J., Remark on well-posedness for the fourth order nonlinear Schr¨odinger type equa-tion, Proc. Amer. Math. Soc. 132 (2004), 3559–3568.[35] Segata, J., Modified wave operators for the fourth-order non-linear Schr¨odinger-type equationwith cubic non-linearity Math. Meth. in the Appl. Sci. 26 No 15 (2006) 1785–1800.[36] Tao, T., Nonlinear dispersive equations, local and global analysis. CBMS. Regional Confer-ence Series in Mathematics, 106. Published for the Conference Boardof the MathematicalScience, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.ISBN: 0-8218-4143-2.[37] Tao, T., Visan, M., and Zhang, X., Minimal-mass blow up solutions of the mass critical NLS. Forum Mathematicum to appear.[38] Thomann, L., Geometric and projective instability for the Gross-Pitaevski equation. Asymp-tot. Anal. 51 (2007), No 3-4, 271–287.[39] Thomann, L., Instabilities for supercritical Schr¨odinger equations in analytic manifolds. J.Diff. Equ. 245 (2008), No 1, 249–280.[40] Visan, M., The defocusing energy-critical nonlinear Schrodinger equation in higher dimensions Duke Math. J. 138 (2007), 281–374. Department of Mathematics, University of Cergy-Pontoise, CNRS UMR 8088, 2, av-enue Adolphe Chauvin, 95302 CERGY-PONTOISE cedex, France E-mail address ::