A CLASS OF SOLUTIONS TO THE 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION THAT BLOW-UP ON A CIRCLE

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1 A CLASS OF SOLUTIONS TO THE 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION THAT BLOW-UP ON A CIRCLE JUSTIN HOLMER AND SVETLANA ROUDENKO Astract. We consider the 3d cuic focusing nonlinear Schrödinger equation (NLS) i t u + u + u u = 0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H axial (R3 ) of initial data, that low-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël s construction [33] of a family of solutions to the d quintic focusing NLS, i t u+ u+ u 4 u = 0, that low-up on a circle.. Introduction Consider, in dimension n, the p-power focusing nonlinear Schrödinger (.) i t u + u + u p u = 0. This equation oeys the scaling symmetry u(x, t) solves (.) = λ p u(λx, λ t) solves (.), which implies that the homogeneous Soolev norm Ḣs is scale invariant provided s = n. The equation (.) also oeys mass, energy, and momentum conservation, p which are respectively defined as M[u] = u L, E[u] = u L p + u p+ L, P [u] = Im u ū dx. p+ In the Ḣ sucritical setting ( < p < + 4 ), there exist soliton solutions u(x, t) = n e it Q(x), where (.) Q + Q + Q p Q = 0. We take Q to e the unique, radial, positive smooth solution (in R n ) of this nonlinear elliptic equation of minimal mass. For further properties see, for example, [4]-[5] and references therein. The local theory in H (p + 4 ) is known from work of Ginire-Velo [0]. Local n existence in time extends to the maximal interval ( T, T ), and if T or T are finite, it is said that the corresponding solution lows up in finite time. The existence of low up solutions are known and the history goes ack to work of Vlasov-Petrishev-Talanov 7 [40], Zakharov 7 [4] and Glassey 77 [] who showed that negative energy solutions with finite variance, xu 0 L <, low up in finite time. Ogawa-Tsutsumi [3]

2 JUSTIN HOLMER AND SVETLANA ROUDENKO extended this result to radial solutions y localizing the variance. Martel [] showed that further relaxation to the nonisotropic finite variance or radiality only in some of the variales guarantees that negative energy solutions low up in finite time. First result for nonradial infinite variance (negative energy) solutions lowing up in finite time was y Glangetas - Merle [] using a concentration-compactness method (see also Nawa [30] for a similar result). Positive energy low up solutions are also known and go ack to [40], [4] (see [38, Theorem 5.] for the precise statement). Turitsyn [39] and Kuznetsov et al [0] extended the low up criteria for finite variance solutions of Ḣ sucritical NLS and, in particular, for the 3d cuic NLS showed that finite variance solutions low up in finite time provided they are under the mass-energy threshold M[u 0 ]E[u 0 ] < c( Q L ), where Q is the ground state solution of (.), and further assumption on the initial size of the gradient u 0 L > c(m[u 0 ], Q L ). Independently, the same conditions for finite variance as well as for radial data (for all Ḣ sucritical, L supercritical NLS) were otained in [4]. [Similar sufficient low up conditions for finite variance or radial data for the Ḣ critical NLS (s = or p = + 4 ) are due to Kenig - Merle [8]; for the situation in the n L critical NLS (s = 0 or p = + 4 ) with n H data refer to Weinstein [4] for the sharp threshold and Merle [] for the characterization of minimal mass low up solutions.] The nonradial infinite variance solutions of the 3d cuic NLS low up in finite or infinite time provided they are under the mass-energy threshold having the same condition on the size of the gradient as discussed aove. This was shown using a variance of the rigidity /concentration-compactness method in [6], thus, extending the result of Glangetas- Merle to positive energy solutions. Further extensions of sufficient conditions for finite time low up were done y Lushnikov [] and Holmer-Platte-Roudenko [3] which include low up solutions aove the mass-energy threshold and are given via variance (or its localized version) and its first derivative if the data is not real. It is also possile to construct solutions which low up in one time direction and scatter or approach the soliton solution (up to the symmetries of the equation) in the other time direction, see [3], [5], [6]. A detailed description of the dynamics of low-up solutions in the L critical (p = + 4 ) case has een developed y Merle-Raphael [5, 8, 4, 7, 6, 3]. They show n that low-up solutions are, to leading order in H, descried y the profile Q (t), rescaled at rate λ(t) ((T t)/ log log(t t) ) /, where (t) / log log(t t). Here, Q is a slight modification of Q see notational item N5 in 3 for the details of the definition in the d cuic case. In the L supercritical, Ḣ sucritical regime ( + 4 < p < + 4 ), one has a n n large family of low-up solutions as discussed aove, ut there are fewer results characterizing the dynamical ehavior of low-up solutions we are only aware of three: Raphaël [33] (quintic NLS in d) and the extension y Raphaël-Szeftel [34] (quintic NLS in all dimensions) and Merle-Raphaël-Szeftel [9] (slighly mass-supercritical

3 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 3 NLS). In this paper, we consider the 3d cuic equation (n = 3, p = 3) which is a physically relevant case in condensed matter and plasma physics. We adapt the method introduced y Raphaël [33] to give a construction of a family of finite-time low-up solutions that low-up on a circle in the xy plane. Raphaël constructed a family of finite-time low-up solutions to the d quintic equation (n =, p = 5) which is Ḣ/ critical, that low-up on a circle. He accomplished this y introducing a radial symmetry assumption, and at the focal point of low-up (without loss r = ), the equation effectively reduces to the d quintic NLS, which is L critical and for which there is a well-developed theory characterizing the dynamics of low-up. We employ a similar dimensional reduction scheme starting from the 3d cuic NLS (which is Ḣ / critical) we impose an axial symmetry assumption, and construct low-up solutions with a focal point at (r, z) = (, 0), where the equation in (r, z) coordinates effectively reduces to the d cuic NLS equation, which is L critical. Our main result is Theorem.. Let Q = Q ( r, z) e as defined in N5 in 3. There exists an open set P in H axial (R3 ), defined precisely in 4, such that the following holds true. Let u 0 P. Then the corresponding solution u(t) to 3d cuic NLS ( (.) with n = 3, p = 3) lows-up in finite time 0 < T < according to the following dynamics. () Description of the singularity formation. There exists λ(t) > 0, r(t) > 0, z(t), and γ(t) R such that, if we define u core (r, z, t) = ( r r(t) λ(t) Q, z z(t) ) e iγ(t), λ(t) λ(t) then ũ(t) = u(t) u core (t), (.3) ũ(t) u in L (R 3 ) as t T, (.4) ũ(t) L (R 3 ) u core (t) L (R 3 ) and the position of the singular circle converges: log(t t) c, (.5) (.6) r(t) r(t ) > 0 as t T, z(t) z(t ) as t T. A result similar to the one presented in this paper was simultaneously developed y Zwiers [43], see remarks at the end of the introduction.

4 4 JUSTIN HOLMER AND SVETLANA ROUDENKO () Estimate on the low-up speed. We have, as t T, (.7) ( ) / T t λ(t), log log(t t) (.8) (t) log log(t t), (.9) γ(t) log(t t) log log(t t). (3) Structure of the L remainder u. For all R > 0 small enough, (.0) u (r) dr dz r r(t ) + z z(t ) R (log log R ), and, in particular, u / L p for p >. (4) H / gain of regularity outside the singular circle. For any R > 0, (.) u H / ( r r(t ) + z z(t ) > R ). A key ingredient in exploiting the cylindrical geometry away from the z-axis is the axially symmetric Gagliardo-Nirenerg inequality, which we prove in. This takes the role of the radial Gagliardo-Nirenerg inequality of Strauss [36] employed y Raphaël. In 3, we collect most of the notation employed, and in 5, we outline the structure of the proof of Theorem.. Most of the argument is a lengthy ootstrap, and in 5, we enumerate the ootstrap input statements (BSI 8) and the corresponding ootstrap output statements (BSO 8). As the output statements are stronger than the input statements, we conclude that all the BSO assertions hold for the full time interval of existence. The steps involved in deducing BSO 8 under the assumptions BSI 8 are outlined in 5, and carried out in detail in 7. already proves (.) in Theorem., the H / gain of regularity outside the singular circle. The proof of Theorem. is completed in 4. In, we prove the log-log rate of low-up (.7). In 3, we prove the convergence of the position of the singular circle, (.5) and (.6). The proof of the size estimates on the remainder profile, (.0) is the same as for Theorem 3 in [6] (we refer the reader to Sections 3, 4, 5 there). Finally, in 4, we prove the convergence of the remainder in L, the estimate (.3). A similar result to Theorem. ut for a slightly smaller class of initial data was recently otained y Zwiers [43] (our Haxial class of initial data is replaced y a smoother version Haxial 3 ). We mention that the methods to prove our main theorem do not treat the cuic equation in higher dimensions, ut it is addressed y Zwiers in [43, Theorem.3]. We think that it should e possile to adapt our method to treat the Ḣ/ critical (p = + 4/(n )) case in higher dimensions, proving the existence of solutions lowing up on a circle (dimension low-up set), however, dealing with fractional nonlinearities is a delicate matter. Zwiers result treats low up sets of codimension in all dimensions; our approach might e ale to treat the low up

5 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 5 sets of dimension in all dimensions. (In 3 dimensions, of course, codimension equals dimension, and thus, the results intersect.).. Acknowledgements. J.H. is partially supported y a Sloan fellowship and NSF grant DMS S.R. is partially supported y NSF grant DMS This project has started at the MSRI program Nonlinear Dispersive Equations in Fall 005 and oth authors are grateful for the support and stimulating workshops during that program. We thank Ian Zwiers and Farice Planchon for remarks on this paper.. A Gagliardo-Nirenerg inequality for axially symmetric functions We egin with an axially symmetric Gagliardo-Nirenerg estimate, analogous to the radial Gagliardo-Nirenerg estimate of Strauss [36]. Consider a function f = f(x, y, z) = f(r cos θ, r sin θ, z) = f(r, z), independent of θ. We call such a function axially symmetric. Lemma.. Suppose that f is axially symmetric. Then for each ɛ > 0, (.) f 4 L 4 (r>ɛ) ɛ f L xyz(r>ɛ) f L xyz(r>ɛ). Proof. The proof is modeled on the classical proof of the Soolev estimates. We use the notation = ( x, y, z ) (i.e., not ( r, z )). Note that f(x, y, z) L xyz = f(r, z) L rdrdz, and also note r f(r, z) = ( x f)(r, z) cos θ + ( y f)(r, z) sin θ, and thus, r f f. Oserve that for a fixed r > ɛ, z R, y the fundamental theorem of calculus and the Cauchy-Schwarz inequality, and also, f(r, z) f(r, z) = f(+, z) f(r, z) + z = ɛ + r =ɛ + r =ɛ f(r, z) r f(r, z) dr f(r, z) r f(r, z) r dr ɛ f(r, z) L r dr (r >ɛ) f(r, z) L r dr (r >ɛ), f(r, z ) z f(r, z ) dz f(r, z ) L z f(r, z ) L z. By multiplying the aove two inequalities, and then integrating against rdrdz, we get f(r, z) 4 r dr dz 4 ( ) f(r, z) r>ɛ,z ɛ L >ɛ) f(r, z) r z dr L (r r dr (r >ɛ) dz ( + ) f(r, z ) L f(r, z ) z L r dr. z r=ɛ Following through with Cauchy-Schwarz in each of the two integrals gives (.).

6 6 JUSTIN HOLMER AND SVETLANA ROUDENKO As a corollary of (.), we have for p 3, that (.) f p+ L p+ p xyz (r>ɛ) ɛ p f L xyz (r>ɛ) f p L xyz (r>ɛ). This follows y the interpolation estimate (Hölder s inequality) f p+ L f 3 p p+ L f p L. 4 Before proceeding, we present a simple application of Lemma.. Corollary.. If u is a cylindrically symmetric solution to i t u + u + u p u = 0 for p < 3 in R 3 that lows-up at finite time T > 0, then low-up must occur along the z-axis. Specifically for any fixed ɛ > 0, (.3) lim t T u L xyz (r<ɛ) = +. Proof. Fix any ɛ > 0. All L p norms will e with respect to dxdydz. For any t > 0, y (.), we have for u = u(t) u L = E + p + u p+ L p+ (r<ɛ) + p + u p+ L p+ (r>ɛ) E + p + u p+ L p+ (r<ɛ) + C ɛ p Using the inequality αβ 3 p α 3 p + p β p, we otain u L 4E + 4 p + u p+ L p+ (r<ɛ) + u L u p L. C ɛ p 3 p u 4 3 p L. Since lim t + u(t) L = +, we otain that lim t + u(t) L p+ (r>ɛ) = +. By the (standard) Gagliardo-Nirenerg inequality, we otain (.3). 3. Notation Recall that we will impose the axial symmetry assumption, i.e., ũ(r, θ, z) = u(r cos θ, r sin θ, z) is assumed independent of θ. The equation (.) in cylindrical coordinates (r, z), assuming the axial symmetry, is (3.) i t u + r ru + r u + zu + u u = 0. Denote y Q = Q(r), r = x, x R, a ground state solution to the d nonlinear elliptic equation (which corresponds to the mass-critical cuic NLS equation in d): Q + R Q + Q Q = 0. We emphasize that Q is a two-dimensional oject.

7 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 7 We now enumerate our notational conventions. N. We will adopt the convention that = ( x, y, z ) is the full gradient and = x + y + z is the full Laplacian. When viewing an axially symmetry function as a function of r and z, we will write the corresponding operators as (r,z) = ( r, z ) and (r,z) = r + z. Note that under the axial symmetry assumption, = (r,z) +r r. N. In the argument, parameters λ(t), γ(t), r(t) and z(t) emerge. Rescaled time, for given 0, is (3.) s(t) = and rescaled position is t 0 r(t) = r r(t) λ(t) dt λ (t ) + s 0, s 0 = e 3π 4 0,, z(t) = z z(t). λ(t) We note that the r and z can oth e negative, although there is the restriction that λ(t) r + r(t) 0. Introduce the full (rescaled) radial variale R = r + z. We have r = λ(t) r + r(t). To help avoid confusion etween r and r, we will use the following notation: given two parameters λ(t) and r(t), define µ λ(t),r(t) ( r) = (λ(t) r + r(t)) {λ(t) r+r(t) 0}. Often the λ(t), r(t) suscript will e dropped. N 3. The inner product (, ) will mean d real inner product in the r, z variales. N 4. If f = f( r, z), then define Λf = f + ( r, z) ( r, z )f, the generator for scaling. Oserve that Λf = f + R Rf. N 5. For a parameter, Q ( r, z) is the d localized self-similar profile. Specifically, following Prop. 8 in [7] which is a refinement of Prop. in [8], given, η > 0, define R = η. There exist universal constants C > 0, η > 0 such that the following holds true. For all 0 < η < η, there exists constants ɛ (η) > 0, (η) > 0 going to zero as η 0 such that for all < (η), there exists a unique radial solution Q (i.e., Q depends only on R) to ( r, z) Q Q + iλq + Q Q = 0 Q ( R)e i R 4 > 0 for R [0, R ) Q ( R ) = 0 Q(0) ɛ (η) < Q (0) < Q(0) + ɛ (η).

8 8 JUSTIN HOLMER AND SVETLANA ROUDENKO N 6. For a parameter, Q ( r, z) = Q ( R) is a truncation of the d localized selfsimilar profile Q. Specifically, (following Prop. 8 in [7]), given > 0, η > 0 small, let R = η R so that R < R. Let φ e a radial smooth cut-off function such that φ (x) = 0 for x R and φ (x) = for x R, and everywhere 0 φ(x), such that φ L + φ L 0 as 0. Now set Then Q ( R) = Q ( R)φ ( R). (3.3) ( r, z) Q Q + iλ Q + Q Q = Ψ, where (3.4) Ψ = ϕ Q + Q ϕ + iq ( r, z) ϕ + (ϕ 3 ϕ )Q Q with the property that for any polynomial P (y) and k = 0, (3.5) P (y) Ψ (k) L e C P /. In terms of Ψ ( r, z), we define an adjusted Ψ (t, r, z) as (3.6) Ψ (t, r, z) = Ψ (t, r, z) so that Q solves λ(t) µ λ(t),r(t) ( r) r Q ( r, z) (3.7) ( r, z) Q + λ µ r Q Q + iλ Q + Q Q = Ψ. We also split Q into real and imaginary parts as Q = Σ + iθ. It is implicitly understood that Σ and Θ depend on (or (t)), and when we want to emphasize dependence, it will e stated explicitly; this decomposition is done only for the truncated profile Q (not for Q ). Similarly, we denote Ψ = Re Ψ + i Im Ψ and Ψ = Re Ψ + i Im Ψ. N 7. The Q satisfies the following properties. QP. ((44) in [8]) Uniform closeness to the ground state. For a fixed universal constant C > 0, e C R( Q Q) C 3 0 as 0. In particular, this implies that e c R k r Θ L ( r, z) 0 as 0 for 0 k 3. QP. ((45) in [8]) Uniform closeness of the derivative Q to the ground state. For a fixed universal constant C > 0, e C R( Q + i R Q) 4 0 as 0. C

9 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 9 QP 3. (Prop. (ii) in [8] and Prop. (iii) in [4]) Degeneracy of the energy and momentum. Specifically, (3.8) E( Q ) e (+Cη)π/, since E( Q ) = Re ΛΨ Q, and (3.9) Im ( r, z) Q Q = 0, Im ( r, z) ( r, z) Q Q = R Q L. QP 4. The profile Q has supercritical mass, and more precisely 0 < d d( ) Q = d 0 < +. =0 QP 5. Algeraic relations corresponding to Galilean, conformal and scaling invariances: (3.0) ( R Q ) R Q + i RΛ Q + R Q Q = R Q RΨ (3.) ( R Q ) R Q + i R Λ Q + R Q Q = 4Λ Q R Ψ (3.) (Λ Q ) Λ Q + (Λ Q ) Q + Q (Σ(ΛΣ) + Θ(ΛΘ)) = ( Q iλ Q Ψ ) ΛΨ iλ Q. The proof of the aove identities are similar to Prop. (iii) in [8] adapted to the d case. For example, to otain the third equation from scaling invariance, multiply (3.7) y λ 3 and take argument to e (λ r, λ z), then differentiate with respect to λ and evaluate the derivative at λ =. Note that d dλ λ= (λ 3 Ψ (λ r, λ z)) = 3Ψ +( r, z) Ψ Ψ + ΛΨ, and thus, we otain the claimed equation. N 8. The linear operator close to Q in (3.7) is M = (M +, M ), where (ɛ = ɛ + iɛ ) M + (ɛ) = ( r, z) ɛ λ ( ) Σ µ rɛ + ɛ Q + Q ɛ ΣΘɛ, M (ɛ) = ( r, z) ɛ λ ( ) Θ µ rɛ + ɛ Q + Q ɛ ΣΘɛ. N 9. For a given parameter, the function ζ ( r, z) is the d linear outgoing radiation. Specifically, following Lemma 5 in [7], there exist universal constants C > 0 and η > 0 such that for all 0 < η < η, there exists (η) > 0 such that for all 0 < < (η), the following holds true: There exists a unique radial solution ζ to ( r + z)ζ ζ + iλζ = Ψ ( r, z )ζ d rd z < +,

10 0 JUSTIN HOLMER AND SVETLANA ROUDENKO where Ψ is the error in the Q equation aove. The numer Γ is defined as the radiative asymptotic, i.e., Γ = lim R ζ ( R). R + The ζ ( r, z) and Γ have the following properties: ZP. Control (and hence, smallness y ZP) of ζ in Ḣ ( r, z) ζ Γ Cη. ZP. Smallness of the radiative asymptotic R > R, e (+Cη)π/ 4 5 Γ R ζ ( R) e ( Cη)π/. N 0. We will make the spectral assumption made in [5, 8, 7, 4, 6]. We note that it involves Q and not Q. A numerically assisted proof is given in Fiich-Merle- Raphaël [9]. Let (see N 4) ΛQ = Q + ( r, z) Q, Λ Q = ΛQ + ( r, z) (ΛQ). Recall that (, ) denotes the d inner product in ( r, z). Consider the two Schrödinger operators L = + 3Q[( r, z) Q], L = + Q[( r, z) Q], and the real-valued quadratic form for ɛ = ɛ + iɛ H : H(ɛ, ɛ) = (L ɛ, ɛ ) + (L ɛ, ɛ ). Then there exists a universal constant δ > 0 such that ɛ H, if the following orthogonality conditions hold: (ɛ, Q) = 0, (ɛ, ΛQ) = 0, (ɛ, yq) = 0, then (ɛ, ΛQ) = 0, (ɛ, Λ Q) = 0, (ɛ, Q) = 0, ( ) H(ɛ, ɛ) δ ɛ + ɛ e y. N. The ring cutoffs are the following. The tight external cutoff is { for 0 r 3 4 and for r 4 3 χ 0 (r) = 0 for 5 r The spectral property was proved in d and in dimensions -4 has a numerically assisted proof.

11 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS The wide external cutoff is { for 0 r and for r 4 4 χ (r) = 0 for r. The internal cutoff is { 0 for 0 r and for r 3 4 ψ(r) = r for r. 4. Description of the set P of initial data We now write down the assumptions on the initial data set P. Let P e the set of axially symmetric u 0 H (R 3 ) of the form u 0 (r, z) = ( r r0 Q0, z z ) 0 e iγ 0 + ũ 0 (r, z), λ 0 λ 0 λ 0 and define the rescaled error with the following controls: ɛ 0 ( r, z) = λ 0 ũ 0 (λ 0 ( r, z) + (r 0, z 0 ))e iγ 0, IDA. Localization of the singular circle: r 0 < α, z 0 < α. IDA. Smallness of 0, or closeness of Q 0 IDA 3. Orthogonality conditions on ɛ 0 : 0 < 0 < α. to Q on the singular circle: (Re ɛ 0, R Σ 0 ) + (Im ɛ 0, R Θ 0 ) = 0, (Re ɛ 0, ( r, z)σ 0 ) + (Im ɛ 0, ( r, z)θ 0 ) = 0, (Re ɛ 0, Λ Θ 0 ) + (Im ɛ 0, Λ Σ 0 ) = 0, (Re ɛ 0, ΛΘ 0 ) + (Im ɛ 0, ΛΣ 0 ) = 0. IDA 4. Smallness condition on ɛ 0 : E(0) ( r, z) ɛ 0 µ λ0,r 0 ( r) d rd z + ɛ 0 ( r, z) e R d rd z Γ ( r, z) 0/ 0 IDA 5. Normalization of the energy and localized momentum (recall ψ from N ): λ 0 E 0 + λ 0 Im ψ u 0 ū 0 Γ 0 0.

12 JUSTIN HOLMER AND SVETLANA ROUDENKO IDA 6. Log-log regime: ( ( )) 8π 0 < λ 0 < exp exp. 9 0 IDA 7. Gloal L smallness: ũ 0 L α. IDA 8. H / smallness outside the singular circle: χ 0 ũ 0 H / α. IDA 9. Closeness to the d ground state mass: u 0 L (dxdydz) Q L (d rd z) + α. IDA 0. Negative energy: IDA. Axial symmetry of u 0. E(u 0 ) < 0. Lemma 4.. The set P is nonempty. Proof. This follows as in Remark 3 of Raphaël [33]. 5. Outline Now let u(t) e the solution to NLS with initial data from the aove set P, and let T > 0 e its maximal time of existence (which at this point could e + ). Because u 0 is axially symmetric and the Laplacian is rotationally invariant in the xy plane, the solution u(t) will e axially symmetric. Thus, we occasionally write u(r, z, t). The first step is to otain a geometrical description of the solution. Since we are not truly in the L critical setting and cannot appeal to the variational characterization of Q, we need to incorporate this geometrical description into the ootstrap argument. The lemma that we need, which follows from the implicit function theorem, is: Lemma 5. (cf. Merle-Raphaël [8] Lemma, [5] Lemma ). If for 0 t t, there exist parameters ( λ(t), γ(t), r(t), z(t), (t)) such that ɛ(t) H α on 0 t t, where ɛ( r, z) = e i γ λu( λ r + r, λ z + z) Q ( r, z), then there exist modified parameters (λ(t), γ(t), r(t), z(t), (t)) such that ɛ defined y ɛ( r, z) = e iγ λu(λ r + r, λ z + z) Q ( r, z) satisfies the following orthogonality conditions (with ɛ = ɛ + iɛ ): ORTH. (ɛ, R Σ (t) ) + (ɛ, R Θ (t) ) = 0 ORTH. (ɛ, ( r, z)σ (t) ) + (ɛ, ( r, z)θ (t) ) = 0

13 ORTH 3. (ɛ, Λ Θ (t) ) + (ɛ, Λ Σ (t) ) = 0 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 3 ORTH 4. (ɛ, ΛΘ (t) ) + (ɛ, ΛΣ (t) ) = 0 and λ(t) u(t) L Q + ɛ(t) H + (t) δ(α 0), L where δ(α 0 ) 0 as α 0 0. Note that the condition ORTH is a vector equation. We also use the notation ũ(r, z, t) = ( r r(t) λ(t) ɛ, z z(t) ), t e iγ(t). λ(t) λ(t) It is important that we consider, y default, u and ũ as 3d ojects in the spatial variales. Thus, when we write ũ L, we mean ũ(x, y, z, t) L (dxdydz) = ũ(r, z, t) L (rdrdz). On the other hand, we consider, y default, Q (t) and ɛ to e d ojects in the spatial variales, and thus, if we were to write Q (t) L, we would just mean Q (t) ( r, z) L (d r,d z). However, if working with the r, z variales and the function Q (t) and ɛ, we will write the integrals out to help avoid confusion. We would like to know that the geometric description holds for 0 t < T, and in addition that properties BSO 8 listed elow hold for all 0 t < T. To show this, we do a ootstrap argument. By IDA and continuity of the flow u(t) in H, we know that for some t > 0, Lemma 5. applies giving (λ(t), γ(t), r(t), z(t), (t)) and ɛ such that ORTH 4 hold on 0 t < t with initial configuration λ(0) = λ 0, etc. Again y the continuity of u(t) (and ɛ(t)) in H we know that BSI 8 hold on 0 t < t y taking t smaller, if necessary. Now take t to e the maximal time for which Lemma 5. applies and BSI 8 hold on 0 t < t. (By the aove reasoning, we must have t > 0.) Under these hypotheses, we show that BSO 8 hold on 0 t < t. Since these properties are all strictly stronger than those of BSI 8, we must have t = T. We now outline this ootstrap argument. We have the following ootstrap inputs that we enumerate as BSI, etc. We assume that all the following properties hold for times 0 t < t < T. BSI. Proximity of r(t) to, or localization of the singular circle r(t) (α ) and proximity of z(t) to 0 z(t) (α ). BSI. Smallness of (t), or closeness of Q to Q on the singular circle 0 < (t) < (α ) 8.

14 4 JUSTIN HOLMER AND SVETLANA ROUDENKO BSI 3. Control of ɛ(t) error y radiative asymptotic Γ (t) E(t) ( r, z) ɛ( r, z, t) µ λ(t),r(t) d rd z + ɛ( r, z, t) e R d rd z Γ 3 4 (t). 0 R < (t) BSI 4. Control of the scaling parameter λ(t) y the radiative asymptotic Γ (t) λ (t) E 0 Γ (t). BSI 5. Control of the r-localized momentum y the radiative asymptotic λ(t) Im ψ u(t) ū(t) dxdydz Γ (t). BSI 6. Control of the scaling parameter λ(t) y (t) ( 0 < λ(t) exp exp BSI 7. Gloal L ound on ũ(t) ũ(t) L (α ) 0. π 0(t) ). BSI 8. Smallness of ũ(t) outside the singular circle χ (r)ũ(t) Ḣ (α ) 4. Assuming that properties BSI 8 hold for 0 t < t < T, we prove that the following ootstrap outputs, laeled BSO 9 hold. BSO. Proximity of r(t) to, or localization of the singular circle and proximity of z(t) to 0 r(t) (α ) 3 z(t) (α ) 3. BSO. Smallness of (t), or closeness of Q to Q on the singular circle 0 < (t) < (α ) 5. BSO 3. Control of ɛ(t) error y radiative asymptotic Γ (t) E(t) ( r, z) ɛ( r, z, t) µ λ(t),r(t) d rd z + ɛ( r, z, t) e R d rd z Γ 4 5 (t). 0 R < (t)

15 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 5 BSO 4. Control of the scaling parameter λ(t) y the radiative asymptotic Γ (t) λ (t) E 0 Γ 4 (t). BSO 5. Control of the r-localized momentum y the radiative asymptotic λ(t) Im ψ u(t) ū(t) dxdydz Γ4 (t). BSO 6. Control of the scaling parameter λ(t) y (t) (which will imply an upper ound on the low-up rate) ( 0 < λ(t) exp exp π ). 5(t) BSO 7. Gloal L ound on ũ(t) ũ(t) L (α ) 5. BSO 8. H smallness of ũ(t) outside the singular circle χ (r)ũ(t) Ḣ (α ) 3 8. The ootstrap argument proceeds in the following steps. The nontrivial steps are detailed in the remaining sections of the paper. Step. Relative sizes of the parameters λ(t), Γ (t), α. Using BSI, BSI 6, and ZP, we have (5.) λ(t) Γ 0 (t). Step. Application of mass conservation. Using BSI, BSI 3, BSI 6, and L conservation, we otain BSO and BSO 7. In other words, mass conservation reinforces the smallness of (t) and also the smallness of the L norm of ũ. This is carried out in 7. Step 3. ɛ interaction energy is dominated y ɛ kinetic energy. That is, the ɛ energy ehaves as if it were L critical and suground state. We otain y splitting the L 4 term in the energy of ɛ into inner and outer radii, using the axial Gagliardo-Nirenerg for outer radii and the usual 3d Gagliardo-Nirenerg for inner radii (5.) ɛ 4 µ( r)d rd z δ(α ) ( r, z) ɛ µ( r) d rd z. This states that in the ɛ-energy, the interaction energy term is suitaly dominated y the kinetic energy term.

16 6 JUSTIN HOLMER AND SVETLANA ROUDENKO A useful statement that comes out of this computation is (5.3) ɛ 4 d rd z ũ L E(t). R<0/ x The proof of this statement only uses the d Gagliardo-Nirenerg inequality (since we have the localization R < 0/) and does not use the H / assumption. This is carried out in 8. Step 4. Energy conservation of u recast as an ɛ statement. Using BSI 3, (5.), BSI, BSI 6, BSI 7, BSI 4, BSI 8, energy conservation, and properties of Q, we otain (ɛ, Σ) + (ɛ, Θ) ( r, z) ɛ µ( r) d rd z (5.4) +3 Q ɛ d rd z + Q ɛ d rd z Γ Cη + δ(α )E(t). R 0/ R <0/ It results from plugging the representation of u in terms of Q (t) and ɛ into the energy conservation equation for u. The result is asically the energy of Q (t), which is small and shows up on the right side as the Γ Cη (t) term, the energy of ɛ, which shows up as the ( r, z) ɛ µ( r) d rd z term on the left (from Step 3, the interaction component of the energy is small and is put on the right), cross terms resulting from u 4 which are linear, quadratic, and cuic in ɛ. The linear terms are kept on the left as (ɛ, Σ) + (ɛ, Θ). The quadratic terms are kept on the left as well, while the cuic term is estimated away. This is carried out in 9. Step 5. Momentum control assumption (BSI 5) recast as an ɛ statement. Using BSI 5, (5.), properties of Q, BSI, BSI, we otain (5.5) (ɛ, ( r, z) Σ) δ(α )E(t) / + Γ. The term that we keep on the left side comes from the cross term. This is carried out in 0. Step 6. Application of the orthogonality conditions. In this step, the orthogonality assumptions are used to deduce laws for time evolution of the parameters λ(t), (t), γ(t). Using the orthogonality conditions ORTH -4, BSI 3, and (5.), we otain the following estimates on the modulation parameters: (5.6) λ s λ + + s c E(t) + Γ Cη,

17 (5.7) CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 7 γ s (ɛ ΛQ, L + (Λ Q)) + (r s, z s ) L λ δ(α )E(t) / + Γ Cη where, we recall, L + = + 3Q. This is carried out in., Step 7. Deduction of BSO 4 from (5.6). Using (5.6), BSI 4, BSI 6, properties of Γ, IDA 3, we otain d ds (λ e 5π/ ) 0, which, upon integrating in time, gives BSO 4. This is carried out in. Step 8. Deduction of a local virial inequality. Using (5.4), (5.6), (5.7), the coercivity property, and the spectral property for d =, the orthogonality conditions, we otain a local virial inequality (5.8) s δ 0 E(t) Γ Cη (t), This is carried out in 3. Step 9. Lower ound on (s) ( = upper ound on low-up rate) Using (5.8), (5.6), BSI 4, BSI 6, BSI, and IDA, IDA 4, we otain (5.9) (s) 3π 4 log s (5.0) λ(s) λ 0 e π 3 which, together imply BSO 6: s log s (5.9) + (5.0) = BSO 6 π 5 (t) log log λ(t). This will later e used to give an upper ound on the low-up rate. (5.9) (5.0) are consequences of a careful integration of (5.8) and an application of the law for the scaling parameter (5.6). This is carried out in 4. Step 0. Control on the radius of concentration. Using (5.7), (5.0), IDA, IDA 4, IDA, we otain BSO. This is carried out in 5. Step. Momentum conservation implies momentum control estimate (BSO 5). Using BSI 3, BSI 7, proof of BSO 6, (5.6), BSI 4, BSI 6, IDA 3, we otain BSO 5. This is carried out in 6. Step. Refined virial inequality in the radiative regime. Here we prove a refinement of (5.8) in the radiative regime. Let φ 3 ( R) = for R and φ 3 ( R) = 0 for R e a radial cutoff. With ζ = φ 3 ( R/A)ζ, A = e a/ for some small constant 0 < a, we define ɛ = ɛ ζ. In this step we estalish (5.) s f (s) δ Ẽ(t) + cγ ɛ d rd z, δ A R A

18 8 JUSTIN HOLMER AND SVETLANA ROUDENKO where f (s) 4 R Q L + r z Im ( r, z) ζ ζ + (ɛ, Λ ζ re ) (ɛ, Λ ζ im ) and Ẽ(t) = This is carried out in 7. ( r, z) ɛ( r, z, t) µ λ(t),r(t) d rd z + 0 R < (t) ɛ( r, z, t) e R d rd z. Step 3. L dispersion at infinity in space. Let φ 4 ( R) e a (nonstrictly) increasing radial cutoff to large radii. Specifically, we require that φ 4 ( R) = 0 for 0 R and φ 4 ( R) = for R 3, with 4 φ 4( R) for R. We next prove, via a flux type computation, an estimate giving control on the term A R A ɛ d rd z: ( (5.) s r(s) φ 4 ( R A This is carried out in 8. ) ɛ µ( r)d rd z ) ɛ Γ a/ 400 A R A E(t) Γ. Step 4. Lyapunov functional in H. By comining (5.) and (5.), we define a Lyapunov functional J and show ) (5.3) s J c (Γ + Ẽ(t) + ɛ, A R A where J (defined later) can e shown to satisfy (5.4) J d 0 δ for some universal constant 0 < δ, and a more refined control { Γ Ca + C (5.5) J (s) f ((s)) E(s) +Γ Ca + CA E(s) with f given y ( f () = and Q Q ) f () = 4 R Q L + Im δ ( 800 f () 0 ( r, z) ζ ζd rd z. ) f (v)dv Here, 0 < df d =0 < +, and hence, (5.5) refines (5.4). This is carried out in 9. Step 5. Deduction of control on E(t) and upper ound on ( = lower ound on low-up rate). By integrating (5.3) and applying (5.4), we prove (5.6) (s) 4π 3 log s

19 and (5.7) CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 9 s s 0 (Γ (σ) + Ẽ(σ) ) dσ c α. Using (5.3) and (5.5), we prove BSO 3, the key dispersive control on the remainder term ɛ. This is carried out in 0. Step 6. H / interior smallness. Using a local smoothing estimate and (5.7), we prove BSO 8. This is carried out in. This concludes the outline of the ootstrap argument, and we now know that BSO 8 hold for all times 0 < t < T. The remainder of the proof of Theorem. is carried out in 4. Recall (3.) and write (6.) u(t, r, z) = ( λ(t) Q r r(t) (t) λ(t) 6. The equation for ɛ, z z(t) ) e iγ(t) + ( r r(t) λ(t) λ(t) ɛ λ(t) In the remainder of this section, Q will always mean Q (t). Recall and thus, r = r r(s) λ(s), z = z z(s), λ(s) s r = r s λ r λ s λ, s z = z s λ z λ s λ. Direct computation when sustituting (6.) into (3.) gives e iγ λ 3 i t u = e iγ λi s u = i λ ( s λ Q + i r s ( λs = i where γ(s) = γ(s) s. Also, and, z z(t) ), t e iγ(t). λ(t) λ r λ ) ( s r Q + i z s λ λ z λ ) s z Q + i s Q γ s Q λ + i s ɛ i λ ( s λ ɛ + i r s λ r λ ) ( s r ɛ + i z s λ λ z λ ) s z ɛ γ s ɛ λ ) λ + ΛQ i λ (r s, z s ) ( r, z) Q + i s Q γ s Q + (iλq Q) ( ) λs + i s ɛ i λ + Λɛ i λ (r s, z s ) ( r, z) ɛ γ s ɛ + (iλɛ ɛ), λ 3 e iγ (r,z) u = ( r, z) Q + ( r, z) ɛ λ 3 e iγ r ru = λ µ rq + λ µ rɛ.

20 0 JUSTIN HOLMER AND SVETLANA ROUDENKO Finally the nonlinear term: λ 3 e iγ u u = Q + ɛ (Q + ɛ) = Q Q + (Re Q ɛ)q + Q ɛ + Q ɛ + (Re Q ɛ)ɛ + ɛ ɛ }{{}}{{}}{{} linear quadratic cuic = Q Q + [(Σ + Q )ɛ + ΣΘɛ ] + i[(θ + Q )ɛ + ΣΘɛ ] + R (ɛ) + ir (ɛ), where the quadratic and cuic terms are put into R(ɛ). Adding up all of the aove in (3.), and taking the real part, we get: ( ) λs (6.) s Θ + s ɛ + M + (ɛ) + Λɛ = λ + ΛΘ + λ (r s, z s ) ( r, z) Θ γ s Σ ( ) λs + λ + Λɛ + λ (r s, z s ) ( r, z) ɛ γ s ɛ Re Ψ + R (ɛ), where (6.3) M + (ɛ) = ɛ ( r, z) ɛ λ ( ) Σ µ rɛ Q + Q ɛ ΣΘɛ. Taking the imaginary part, we get (6.4) s Σ + s ɛ M (ɛ) + Λɛ = ( λs ) λ + ΛΣ + λ (r s, z s ) ( r, z) Σ + γ s Θ ( ) λs + λ + Λɛ + λ (r s, z s ) ( r, z) ɛ + γ s ɛ + Im Ψ R (ɛ), where (6.5) M (ɛ) = ɛ ( r, z) ɛ λ ( ) Θ µ rɛ Q + Q ɛ ΣΘɛ. 7. Bootstrap Step. Application of mass conservation In this section we prove BSO and BSO 7 are consequences of the L conservation of u(t) and several other BSI s. The assumed smallness of (t), ɛ(t), and λ(t), initial smallness of ũ (at t = 0) comined with L norm conservation for u(t) reinforces the smallness of ũ(t) in L for 0 < t < t and smallness of (t) for 0 < t < t. Recall (6.) and denote ũ(r, z, t) = λ(t) ɛ ( r r(t) λ(t), z z(t) ), t e iγ(t). λ(t)

21 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS Sustitute (6.) into the mass conservation law u(r, z, t) rdrdz = u 0 L and rescale to otain (7.) u 0 L = Q (t) ( r, z) µ( r) d rd z + Re Q (t) ( r, z) ɛ( r, z) µ( r) d rd z + ũ(r, z, t) rdrdz. In the first term, write out µ( r) = λ r + (r(t) ) +, which splits the integral into three pieces. The third piece we keep; the first two we estimate using [BSI 6 : λ exp( exp π/0(t)) and BSI : (t) (α ) /8 ] = λ (α ) / and BSI : r(t) (α ) /. The second term in (7.) we estimate using Cauchy- Schwarz and the assumed control on ɛ(t) given y BSI 3, ZP, and BSI : E(t) Γ 3/4 (t) (α ) /. Collecting the results, we now have Q (t) d rd z + ũ L = u 0 L + O((α ) / ). Finally, we use QP3: which gives d( ) d d Q L, d r z 0 = d(0) > 0, d differentiale, Q L r z Since 4 (α ) /, we have and conclude y IDA 9 to get = Q L r z + d 0 + O( 4 sup 0 σ d (σ) ). d 0 (t) + ũ(t) L xyz = u 0 L Q L r z + O((α ) / ) which implies BSO and BSO 7. d 0 (t) + ũ(t) L xyz c(α ) /, 8. Bootstrap Step 3. Interaction energy kinetic energy for ɛ Here we deduce (5.) (or (3.94) in [33]), which controls ɛ 4 y ɛ, and is useful later on. By change of variales, we have ɛ 4 µ( r) d rd z = λ (t) ũ 4 r drdz + λ (t) ũ 4 r drdz. r r

22 JUSTIN HOLMER AND SVETLANA ROUDENKO For the first term, we use the (r-localized) 3d Gagliardo-Nirenerg estimate. For the second term, we use the axially symmetric exterior Gagliardo-Nirenerg estimate Lemma.. Together, they give the ound λ (t) ũ H {xyz, r < 4 } ũ L xyz + λ (t) ũ L {xyz, r > 4 } ũ L xyz. We then apply BSI 8 (smallness of ũ Ḣ/ for r < 4 ) and BSO 7 (smallness of ũ L gloally in r) to get the estimate (α ) /4 λ (t) xyz ũ L xyz. For axially symmetric functions, we have r u = x u + y u, so we may replace xyz y rz, then rescale to otain and this is (5.). (α ) /4 r, z ( r, z) ɛ µ( r) d rd z 9. Bootstrap Step 4. Energy conservation In this step, we prove (5.4) as a consequence of various ootstrap assumptions and the conservation of energy. Plug (6.) into the energy conservation identity λ E 0 = λ (r,z) u rdrdz λ u 4 rdrdz. The result for the first term on the right side is λ (r,z) u rdrdz = ( r, z) Q(t) µ( r) d rd z + Re ( r, z) Q(t) ( r, z) ɛ µ( r)d rd z + ( r, z) ɛ µ( r)d rd z = I. + I. + I.3.

23 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 3 For the second term, we otain λ u 4 rdrdz = (t) 4 µ( r) d rd z Q (t) Re( Q (t) ɛ)µ( r) d rd z (Re Q (t) ɛ) µ( r) d rd z Q (t) ɛ µ( r) d rd z quadratic ɛ Re( Q (t) ɛ)µ( r) d rd z cuic ɛ 4 µ( r) d rd z = II. + II. + II.3 + II.4 + II.5 + II.6. In all of these terms except I.3 and II.6, we can write out µ( r) = λ r + r(t) and discard the λ r term y estimation, since λ(t) Γ 0 (t). We should thus hereafter in this section replace µ( r) with r(t) for all terms ut I.3 and II.6. We have the Q (t) energy terms: r(t) (I. + II.) = E( Q (t) ). We also have the linear in ɛ terms that we comine, and then sustitute the equation for Q (t) to get r(t) (I. + II.) = Re ( ( r, z) Q(t) + Q (t) Q(t) ) ɛ d rd z = Re Q (t) ɛ d rd z Im Λ Q (t) ɛ d rd z + Re Ψ (t) ɛ d rd z. The middle term is zero y ORTH 4, and the first term can e rewritten to otain r(t) (I. + II.) = (Σ, ɛ ) (Θ, ɛ ) + Re Ψ (t) ɛ d rd z. Next, for the quadratic in ɛ terms in II, replace Q (t) y Q and use the proximity estimates for Q (t) to Q to control the error r(t) (II.3 + II.4) = 3 Q ɛ d rd z Q ɛ d rd z + errors. For the cuic term in II, we estimate as (using that / /λ) r(t) II.5 ɛ 3 Q (t) d rd z ( ) / ( ɛ 4 d rd z R /0λ ɛ Q d rd z) /.

24 4 JUSTIN HOLMER AND SVETLANA ROUDENKO By d Gagliardo-Nirenerg applied to the first term (where φ is a smooth cutoff to R /0λ) ( ) / ( ɛ d rd z R /0λ ) / ( ( r, z) [φ( Rλ)ɛ( R)] d rd z ɛ Q d rd z) /. Note that if R /0λ, then µ( r). The first term aove is controlled y some α power (using the R restriction to reinsert a µ( r) factor) y rescaling ack to ũ y BSI 7. The second term is controlled as ( ) / λ ɛ d r d z + ɛ d r d z R λ λ ũ L + ( λ + E(t) /. R λ R λ ɛ µ( r) d r d z ) / For the third term, we use Q (t) e R/. These considerations give r(t) II.5 (α ) /5 (λ + E(t)). The quartic term in ɛ in II is controlled y (5.). We next note that BSI : r(t) (α ) / = r(t). Collecting all of the aove estimates and manipulations, we otain (5.4). 0. Bootstrap Step 5. Momentum control assumption recast as an ɛ statement In this section we prove (5.5). Recall BSI 5: λ(t) Im ψ u(t) ū(t) dxdydz Γ (t). A asic calculus fact states that for axially symmetric functions f and g, (x,y,z) f (x,y,z) g = (r,z) f (r,z) g. Plug in (6.) into the left side of BSI 5 and change variales (r, z) to ( r, z) to get λ(t) Im ψ u(t) ū(t) dxdydz = Im [ ( (r,z) ψ)(r, z) ( ) ] ( r, z) ( Q (t) + ɛ) ( r, z) ( Q (t) + ɛ)( r, z) µ( r)d r d z,

25 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 5 then using that ψ(r) = (, 0) on the support of Q (t), we continue as = Im r Q(t) Q(t) µ( r) d rd z + Im r Q(t) ɛµ( r) d rd z + Im r ɛ ɛ µ( r) d rd z = I + II + III. In term I, we expand out µ( r) = λ(t) r + r(t) which gives two terms: the first of these we estimate out and use λ(t) Γ (t), and the second of these is 0 y QP 3. In term II, we expand out µ( r) to get (0.) II = λ(t) Im r Q(t) ɛ r d r d z + r(t)( r Θ, ɛ ) r(t)( r Σ, ɛ ). The first of these is estimated away using λ(t) Γ (t), the third term we keep, and for the second we note e R/ r Θ L ( r, z) 0 as 0 y QP, and thus, we can estimate ( r Θ, ɛ ) y Cauchy-Schwarz. For term III, we first estimate y Cauchy-Schwarz to get ( ) / ( / III r ɛ µ( r) d rd z ɛ µ( r) d rd z). For the second factor, we convert ɛ ack in terms of ũ: ɛ( r, z) = λ(t)ũ(λ(t) r + r(t), λ(t) z + z(t), t) to otain ɛ µ( r) d rd z = ũ L, which is α /5 y BSO. Comining all of the aove information, we get (0.) r(t)( r Σ, ɛ ) δ(α )E(t) / + Γ (t). We finish y using that and then (5.5) follows. BSI : r(t) (α ) / = r(t)

26 6 JUSTIN HOLMER AND SVETLANA ROUDENKO. Bootstrap Step 6. Application of orthogonality conditions.. Computation of λ s /λ +. Here we explain how to otain (5.6). Multiply the equation for ɛ (6.4) y R Σ and the equation for ɛ (6.) y R Θ and add. We study the resulting terms one y one. Term. (.) s [( Σ, R Σ) + ( Θ, R ] Θ) = s Re( Q, R Q ). Using the QP properties: e R/ ( Q i R Q) 4 0 as 0 C and e R/ ( Q Q) C 3 0 as 0 to show that (.) s Re( 4 i R Q, R Q) = 0. More specifically, we can show that (.) is s δ(), where δ() 0 as 0. Term. Using ORTH condition, we have (.) ( s ɛ, R Σ) + ( s ɛ, R Θ) = s [(ɛ, R Σ) + (ɛ, R ] [ Θ) s (ɛ, R Σ) + (ɛ, R ] Θ) = s [(ɛ, R Σ) + (ɛ, R ] Θ), which we then estimate as ( s R 0 ɛ e Rd rd z) / s E(t) /. Term 3. (.3) ( M (ɛ) + Λɛ, R Σ) + (M + (ɛ) + Λɛ, R Θ) = (ɛ, R Σ) + (ɛ, R Θ) + ( ( r, z) ɛ, R Σ) ( ( r, z) ɛ, R Θ) + λ µ ( rɛ, R Σ) λ µ ( rɛ, R Θ) (( ) ) (( ) ) Θ + Q + Q ɛ, R Σ Σ Q + Q ɛ, R Θ + (ΣΘɛ, R Σ) (ΣΘɛ, R Θ) + [(Λɛ, R Σ) + (Λɛ, R Θ)] = I + II + III + IV + V + VI.

27 By integration y parts, CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 7 I + II = (ɛ, R Σ) + (ɛ, R Θ) + (ɛ, ( r, z) ( R Σ)) (ɛ, ( r, z) ( R Θ)). Computation gives, for any function f, that ( r, z) ( R f) = 4Λf + R ( r, z) f. Thus, using ORTH 4, I + II = Im(ɛ, R ( Q + Q )). Sustituting the equation for Q gives I + II = Im(ɛ, R ( iλ Q Q Q Ψ )). By examining IV+V and rearranging terms, and thus, IV + V = Im(ɛ, R Q Q ), I + II + IV + V = Im(ɛ, R ( iλ Q Ψ )). In fact, adding VI cancels the middle term: Properties (3.5) and ZP imply I + II + IV + V + VI = Im(ɛ, R Ψ )). (.4) P (y)ψ (k) (y) L C P,k Γ ( Cη), thus, we get (ɛ, R Ψ ) Γ ( Cη) For III, use λ Γ 0 Together we have Term 4. (.5) ( ) / ɛ e R d rd z = Γ ( Cη) E(t) /. R<0/ and µ and Cauchy-Schwarz to otain ( III Γ 0 / ɛ e d rd z) R. R<0/ ( Rd rd z) / (.3) Γ ( Cη) ɛ e. R<0/ ( ) λs λ + [(ΛΣ, R Σ) + (ΛΘ, R Θ)] By integration y parts, for any function f, we have (Λf, R f) = R f, and thus, the aove expression ecomes ( ) λs λ + R Q L.

28 8 JUSTIN HOLMER AND SVETLANA ROUDENKO Term 5. (.6) ( ) ( ) λ (r s, z s ) ( r, z) Σ, R Σ + λ (r s, z s ) ( r, z) Θ, R Θ Each of these two terms, it turns out, is zero. For example, ( r Θ, R Θ) = R r (Θ )d rd z = r Θ d rd z = 0, since it is the integral of an odd function (recall r goes from to + ). Term 6. (.7) γ s [(Θ, R Σ) (Σ, R Θ)] = 0. Term 7. (.8) y Cauchy-Schwarz. Term 8. ( (rs, z s ) (.9) λ we estimate y similar to Term 7. Term 9. ( ) λs λ + [(Λɛ, R Θ) + (Λɛ, R Σ)] λ s λ + E(t)/, ) ( ) ( r, z) ɛ, R (rs, z s ) Σ + ( r, z) ɛ, λ R Θ (r s, z s ) E(t) /, λ (.0) γ s [(ɛ, R Σ) (ɛ, R Θ)] γ s E(t) /. Term 0. (.) (Im Ψ, R Σ) (Re Ψ, R Θ) = Im( Ψ, R Q ). It turns out that this term is merely zero ( E(t)), which is shown y sustituting the equation for Ψ in terms of Q into (.) and applying the property (3.9), i.e., Im ( r, z) ( r, z) Q Q d rd z + R Q L = 0 and Term. λ µ ( r Q, R Q ) Γ 0. (.) (R (ɛ), R Θ) (R (ɛ), R Σ)

29 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 9 Recall that R (ɛ) and R (ɛ) consist of quadratic and cuic terms. A typical quadratic term has the form ( ) ɛ Q d rd z ɛ e R d rd z e + R/ Q L E(t). R<0/ For the cuic terms, we use Cauchy-Schwarz, (5.3) and properties of Q. A typical term is ( ) / ( ) / ɛ 3 Q d rd z ɛ 4 d rd z ɛ Q R<0/ ( ) / ( R) / ɛ 4 d rd z ɛ e e + R/ Q L R<0/ E(t). Collecting the aove estimates on terms (.) (.), keeping only (.5), we otain (.3) ( λ s λ + δ() s + E(t) / λ s λ + + (r ) s, z s ) + γ s + E(t) / Γ ( Cη) + E(t). λ.. Computation of (r s, z s )/λ. Multiply the equation for ɛ (6.4) y ( r, z)σ and the equation for ɛ (6.) y ( r, z)θ and add. Note we will have now a vectorial equation and again study each term separately. Term. (.4) s [( Σ, ( r, z)σ) + ( Θ, ( r, z)θ)] = s Re( Q, ( r, z) Q ). Recall that Q i R Q, and 4 e R/ ( Q Q) 0 as 0, hence, we have C 3 s Re( Q, ( r, z) Q ) Re( i 4 R Q, ( r, z)q) = 0, or similar to Term in (.), s δ() with δ() 0 as 0. Term. Using ORTH condition, we have (.5) ( s ɛ, ( r, z)σ) + ( s ɛ, ( r, z)θ) = s [(ɛ, ( r, z)σ) + (ɛ, ( r, z)θ)] s [(ɛ, ( r, z) Σ) + (ɛ, ( r, z) Θ)] = s Re(ɛ, ( r, z) Q ) s δ(), since again Q i 4 R Q and Re(ɛ, ( r, z) Q ) 0.

30 30 JUSTIN HOLMER AND SVETLANA ROUDENKO Term 3. ( M (ɛ) + Λɛ, ( r, z)σ) + (M + (ɛ) + Λɛ, ( r, z)θ) = Im(ɛ, ( r, z) Q ) + Im( ɛ, ( r, z) Q ) + λ µ Im( rɛ, ( r, z) Q ) (.6) + Im(ɛ, ( r, z) Q Q ) + Im(ɛ, ( r, z) iλ Q ) = Im(ɛ, ( r, z) [ Q + Q + Q Q + iλ Q ] ) + Im(ɛ, ( r, z) Q ) + λ µ Im( rɛ, ( r, z) Q ) [ = Im (ɛ, ( r, z)ψ ) + (ɛ, ( r, z) Q ) + λ ] µ ( rɛ, ( r, z) Q ) = I + II + III. For term I, we use the estimate (.4) for Ψ to get (ɛ, ( r, z)ψ ) Γ ( Cη) E(t) /. The term II (for example, the first coordinate of the vector with r Q ) we write as [(ɛ, r Σ) (ɛ, r Θ)] + (ɛ, r Σ) (ɛ, r Θ) and note that r Θ 0 as 0, thus, (ɛ, r Θ) δ()e(t) /, and the term (ɛ, r Σ) δ(α )E(t) / + Γ, which we estimated as in (0.). The last two terms are estimated out y Cauchy-Schwarz and properties QP. For III, use λ Γ 0 and µ and Cauchy-Schwarz to otain III Γ 0 E(t) /. Together we have Term 4. (.7) Term 5. (.8) (.3) Γ ( Cη) E(t) / + (δ() + δ(α ))E(t) / + Γ + Γ 0 E(t) /. ( ) ( ) λs λ + λs [(ΛΣ, ( r, z)σ)+(λθ, ( r, z)θ)] = λ + Re(Λ Q, ( r, z) Q ) = 0. ( ) ( ) λ (r s, z s ) ( r, z) Σ, ( r, z)σ + λ (r s, z s ) ( r, z) Θ, ( r, z)θ ( ) = Re λ (r s, z s ) ( r, z) Q, ( r, z) Q = (r s, z s ) λ Q L ( r, z). Term 6. (.9) γ s [(Θ, ( r, z)σ) (Σ, ( r, z)θ)] = 0.

31 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 3 Term 7. (.0) ( ) λs λ + [(Λɛ, ( r, z)θ) + (Λɛ, ( r, z)σ)] λ s λ + E(t)/, applying ORTH and y Cauchy-Schwarz. Term 8. (.) ( (rs, z s ) λ similar to Term 7. Term 9. ) ( ) (rs, z s ) ( r, z) ɛ, ( r, z)σ + ( r, z) ɛ, ( r, z)θ λ (.) γ s [(ɛ, ( r, z)σ) (ɛ, ( r, z)θ)] γ s E(t) /. Term 0. (.3) (Im Ψ, ( r, z)σ) (Re Ψ, ( r, z)θ) = Im( Ψ, ( r, z) Q ). (r s, z s ) E(t) /, λ Sustituting (3.7) for Ψ in terms of Q, one can see that all terms are zero (integration y parts or y degeneracy of the momentum QP 3, first property in (3.9)) except for λ Im( Q µ r, ( r, z) Q ) which is ounded y Γ 0 since λ < Γ 0 (and localization of Q implies µ and oundedness of the inner product). Term. (.4) (R (ɛ), ( r, z)θ) (R (ɛ), ( r, z)σ) E(t), in the same fashion as Term in (.). Collecting the aove estimates on terms (.4) (.4), keeping only (.8), we otain (.5) ( (r s, z s ) λ Q L δ() s +E(t) / λ s λ + + (r ) s, z s ) + γ s +E(t) / Γ ( Cη) +E(t). λ.3. Computation of s. Multiply the equation for ɛ (6.4) y ΛΘ and the equation for ɛ (6.) y ΛΣ and add. Term. (.6) s [( Θ, ΛΣ) ( Σ, ΛΘ)] = s Im( Q, Λ Q ). Recalling that Q i R Q and Λ Q 4 = Q + R R Q, we estimate (.6) as s Im( Q, Λ Q ) 4 R Q L ( r, z) + R 3 Q R Q ) s 4 R Q L ( r, z).

32 3 JUSTIN HOLMER AND SVETLANA ROUDENKO Term. Using ORTH 4 condition, we have ( s ɛ, ΛΣ) ( s ɛ, ΛΘ) = s [(ɛ, ΛΣ) + (ɛ, ΛΘ)] + s [(ɛ, Λ Σ) (ɛ, Λ Θ)] (.7) = s Im(ɛ, Λ Q ) s E(t) /, y Cauchy-Schwarz and using properties of Q (e.g., see (95) in [8]). Term 3. (M + (ɛ) + Λɛ, ΛΣ) + (M (ɛ) Λɛ, ΛΘ) (.8) = Re(ɛ, Λ Q ) Re( ɛ, Λ Q ) λ µ Re( rɛ, Λ Q ) Re(ɛ, Q Λ Q + Q (ΣΛΣ + ΘΛΘ)) + [(Λɛ, ΛΣ) (Λɛ, ΛΘ)] = I + II + III + IV + V. For term III, use λ Γ 0, localization of Q, thus, µ, and Cauchy-Schwarz to otain III Γ 0 E(t) /. For term IV we use ORTH 4 to otain ( IV = [(( r, z) ɛ, ΛΣ) (( r, z) ɛ, ΛΘ)] = Re i (( r, z) ɛ, Λ Q ) ). For terms I, II and IV we use QP 5 scaling invariance property (3.): I + II + IV = Re(ɛ, ( Q iλ Q Ψ ) ΛΨ iλ Q ). Terms with Ψ we estimate y (.4), terms with i we comine with the term IV aove and estimate using BSI (smallness of ), Cauchy-Schwarz and localization properties of Q, thus, Terms 3 is ounded y Γ ( Cη) E(t) /. Term 4. ( ) λs (.9) λ + [(ΛΘ, ΛΣ) (ΛΣ, ΛΘ)] = 0. Term 5. (.30) ( ) ( ) λ (r s, z s ) ( r, z) Θ, ΛΣ λ (r s, z s ) ( r, z) Σ, ΛΘ ( ) = Im λ (r s, z s ) ( r, z) Q, Λ Q = 0, y the degeneracy of the momentum QP 3. Term 6. (.3) γ s [(Σ, ΛΣ) + (Θ, ΛΘ)] = γ s Re( Q, Λ Q ) = 0.

33 CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 33 Term 7. (.3) ( ) ( ) λs λ + λs [(Λɛ, ΛΣ) (Λɛ, ΛΘ)] = λ + Im(Λɛ, Λ Q ) (.33) λ ( ) (rs, z s ) = Im ( r, z) ɛ, Λ λ Q λ s λ + E(t)/, y Cauchy-Schwarz and closeness of Q to Q and properties of Q (as in Term aove). Term 8. ( ) ( ) (rs, z s ) (rs, z s ) ( r, z) ɛ, ΛΣ ( r, z) ɛ, ΛΘ λ Term 9. (r s, z s ) E(t) /. λ (.34) γ s [(ɛ, ΛΣ) + (ɛ, ΛΘ)] γ s E(t) /. Term 0. (.35) (Re Ψ, ΛΣ) + (Im Ψ, ΛΘ) = Re( Ψ, Λ Q ), which is estimated y δ(α ), since Ψ, Λ Q e C/, see Lemma 4 in [8]. Term. (.36) (R (ɛ), ΛΘ) (R (ɛ), ΛΣ) E(t), estimating quadratic and cuic in ɛ terms similar to Term in (.). Collecting the aove estimates on terms (.6) (.36), keeping only (.6), we otain (.37) ( s RQ L λ s E(t)/ λ + + (r ) s, z s ) + γ s + s + E(t) / Γ ( Cη) + E(t). λ.4. Computation of γ s. Multiply the equation for ɛ (6.4) y Λ Θ and the equation for ɛ (6.) y Λ Σ and add. Term. (.38) s [ ( Σ, Λ Θ) ( Θ, Λ Σ) ] = s Im( Q, Λ Q ) δ(α ) y the properties of Q (see the last estimate in Lemma 4 in [8]). Term. Using ORTH 3 condition, we have (.39) ( s ɛ, Λ Θ) ( s ɛ, Λ Σ) = s [ (ɛ, Λ Θ) (ɛ, Λ Σ) ] s [ (ɛ, Λ Θ) (ɛ, Λ Σ) ] = s Im(ɛ, Λ Q ) s E(t) / y the estimate (ɛ, P (R) dm dr m Q (R) E(t) /, 0 m, from Lemma 4 in [8].

34 34 JUSTIN HOLMER AND SVETLANA ROUDENKO Term 3. (.40) ( M (ɛ) + Λɛ, Λ Θ) (M + (ɛ) + Λɛ, Λ Σ) = (ɛ, Λ Σ) (ɛ, Λ Θ) + ( ( r, z) ɛ, Λ Σ) + ( ( r, z) ɛ, Λ Θ) + λ µ ( rɛ, Λ Σ) + λ µ ( rɛ, Λ Θ) ( + (Θ + Q ) ( )ɛ, Λ Θ + (Σ + Q ) )ɛ, Λ Σ + (ΣΘɛ, Λ Σ) + (ΣΘɛ, Λ Θ) + [(Λɛ, Λ Σ) (Λɛ, Λ Θ)] = I + II + III + IV + V + VI. For terms III and VI we use the smallness of λ and, localization of Q and Cauchy-Schwarz to estimate them y Γ 0 E(t) /. In terms I, II, IV and V we collect separately terms containing ɛ and ɛ. Recall the closeness of Q to Q and that Q is real, thus, for example, r Θ 0 as 0 (recall QP ), and hence, the terms containing r Θ will e on the order of δ() - these are the terms containing ɛ. The terms with ɛ produce (ɛ, Λ Σ + (Λ Q ) + Λ Σ( Q + Σ ) + ΣΘΛ Θ) = (ɛ, L + (Λ Q)) + δ(), where L + = + 3Q, y property QP. Term 4. ( ) λs λ + [(ΛΣ, Λ Θ) (ΛΘ, Λ Σ)] ( ) λs (.4) = λ + (ΛΣ, ( r, z) (ΛΘ)) λ s λ + δ(), again y QP, closeness of Q to Q (e.g., the terms such as r Θ 0 as 0). Term 5. ( ) ( ) (.4) λ (r s, z s ) ( r, z) Σ, Λ Θ λ (r s, z s ) ( r, z) Θ, Λ Σ (r s, z s ) δ(), λ y QP similar to Term 4. Term 6. (.43) γ s [(Σ, Λ Σ) + (Θ, Λ Θ)] = γ s ( ΛQ L + δ()), y integration y parts and closeness of Q to Q.

35 Term 7. (.44) CIRCLE BLOW-UP SOLUTIONS TO 3D NLS 35 ( ) λs λ + [(Λɛ, Λ Θ) (Λɛ, Λ Σ)] y Cauchy-Schwarz and properties of Q. Term 8. ( (rs, z s ) (.45) λ similar to Term 7. Term 9. λ s λ + E(t)/, ) ( ) ( r, z) ɛ, Λ (rs, z s ) Θ ( r, z) ɛ, Λ Σ λ (.46) γ s [(ɛ, Λ Θ) + (ɛ, Λ Σ)] γ s E(t) /. Term 0. (.47) (Im Ψ, Λ Θ) + (Re Ψ, Λ Σ) = Re( Ψ, Λ Q ) δ(α ). Term. (.48) (R (ɛ), Λ Σ) (R (ɛ), Λ Θ) E(t), (r s, z s ) E(t) /, λ in the same fashion as Term in (.). Collecting the aove estimates on terms (.38) (.48), keeping only (.43) and the estimate for (.40), we otain γ s (ɛ ΛQ, L + (Λ Q)) L (.49) E(t) ( / s + λ s λ + + (r ) s, z s ) + γ s + E(t) / Γ ( Cη) + E(t). λ We finish this section y oserving that solving the system of equations (.3)- (.5)-(.37)-(.49) for parameters ( s, λs (rs,zs) +,, γ λ λ s ) gives (5.6) and (5.7).. Bootstrap Step 7. Deduction of BSO 4 from (5.6) From BSI 3: E(t) Γ 3/4 (t), we deduce from (5.6) the estimate λ s λ + + s Γ / (t). Direct computation gives (.) d ds (λ e 5π/ ) = λ e 5π/ ( λs λ 5π ) s ( = λ e 5π/ λs λ + 5π s ).