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69
Global wellposedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in R 1+4
, 2006
"... We obtain global wellposedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energyspace solutions to the defocusing energycritical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequencylocalized inte ..."
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Cited by 70 (15 self)
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We obtain global wellposedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energyspace solutions to the defocusing energycritical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequencylocalized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the L6 t,xnorm
Nonlinear Schrödinger equations at critical regularity
 CLAY LECTURE NOTES
, 2009
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Scattering for the nonradial 3d cubic nonlinear Schrödinger equation
"... Abstract. Scattering of radial H1 solutions to the 3D focusing cubic nonlinear Schrödinger equation below a massenergy threshold M[u]E[u] < M[Q]E[Q] and satisfying an initial massgradient bound ‖u0‖L2‖∇u0‖L2 < ‖Q‖L2‖∇Q‖L2, where Q is the ground state, was established in HolmerRoudenko [8]. ..."
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Cited by 37 (6 self)
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Abstract. Scattering of radial H1 solutions to the 3D focusing cubic nonlinear Schrödinger equation below a massenergy threshold M[u]E[u] < M[Q]E[Q] and satisfying an initial massgradient bound ‖u0‖L2‖∇u0‖L2 < ‖Q‖L2‖∇Q‖L2, where Q is the ground state, was established in HolmerRoudenko [8]. In this note, we extend the result in [8] to nonradial H1 data. For this, we prove a nonradial profile decomposition involving a spatial translation parameter. Then, in the spirit of KenigMerle [10], we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned. 1.
Global wellposedness and scattering for the defocusing L²critical nonlinear Schrödinger equation when d = 1
, 2015
"... In this paper we prove global well posedness and scattering for the defocusing, one dimensional mass critical nonlinear Schrödinger equation. We make use of a long time Strichartz estimate and a frequency localized Morawetz estimate. This continues work begun in [28] and [30] for dimensions d ≥ ..."
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Cited by 34 (7 self)
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In this paper we prove global well posedness and scattering for the defocusing, one dimensional mass critical nonlinear Schrödinger equation. We make use of a long time Strichartz estimate and a frequency localized Morawetz estimate. This continues work begun in [28] and [30] for dimensions d ≥ 3 and d = 2 respectively.
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation
 Comm. Math. Phys
"... Abstract. We consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂tu + ∆u + u  2u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expres ..."
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Cited by 34 (6 self)
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Abstract. We consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂tu + ∆u + u  2u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scaleinvariant quantities ‖u0‖L2‖∇u0‖L2 and M[u]E[u], where u0 denotes the initial data, and M[u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution eitQ(x), where Q is the groundstate solution to a nonlinear elliptic equation, and we prove that if M[u]E[u] < M[Q]E[Q] and ‖u0‖L2‖∇u0‖L2 < ‖Q‖L2‖∇Q‖L2, then the solution u(t) is globally wellposed and scatters. This condition is sharp in the sense that the soliton solution eitQ(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if M[u]E[u] < M[Q]E[Q] and ‖u0‖L2‖∇u0‖L2> ‖Q‖L2‖∇Q‖L2, then the solution blowsup in finite time. The technique employed is parallel to that employed by KenigMerle [16] in their study of the energycritical NLS. 1.
A (concentration)compact attractor for highdimensional nonlinear Schrödinger equations, Dyn
 Partial Differ. Equ
"... Abstract. We study the asymptotic behavior of large data solutions to Schrödinger equations iut + ∆u = F(u) in R d, assuming globally bounded H 1 x (Rd) norm (i.e. no blowup in the energy space), in high dimensions d ≥ 5 and with nonlinearity which is energysubcritical and masssupercritical. In th ..."
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Cited by 25 (3 self)
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Abstract. We study the asymptotic behavior of large data solutions to Schrödinger equations iut + ∆u = F(u) in R d, assuming globally bounded H 1 x (Rd) norm (i.e. no blowup in the energy space), in high dimensions d ≥ 5 and with nonlinearity which is energysubcritical and masssupercritical. In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in H 1 x (Rd) to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H 1 x(R d). This is despite the total lack of any dissipation in the equation. This statement can be viewed as weak form of the “soliton resolution conjecture”. We also obtain a more complicated analogue of this result for the nonsphericallysymmetric case. As a corollary we obtain the “petite conjecture ” of Soffer in the high dimensional noncritical case. 1.
Tensor products and correlation estimates with applications to nonlinear Schrödinger equations
"... We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain’s bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the orig ..."
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Cited by 24 (5 self)
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We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain’s bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the GaussWeierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schrödinger equations and derive a direct proof of Nakanishi’s H 1 scattering result for every L²supercritical nonlinearity. We also prove scattering below the energy space for a certain class of L²supercritical equations.
Global wellposedness and scattering for the energycritical, defocusing Hartree equation for radial data
 J. Funct. Anal
"... Using the same induction on energy argument in both frequency space and spatial space simultaneously as in [6], [31] and [35], we obtain global wellposedness and scattering of energy solutions of defocusing energycritical nonlinear Hartree equation in R × R n (n ≥ 5), which removes the radial assu ..."
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Cited by 21 (15 self)
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Using the same induction on energy argument in both frequency space and spatial space simultaneously as in [6], [31] and [35], we obtain global wellposedness and scattering of energy solutions of defocusing energycritical nonlinear Hartree equation in R × R n (n ≥ 5), which removes the radial assumption on the data in [25]. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition 3.1 and Corollary 3.1) of the minimal energy blow up solutions, which can not be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that L 2n n−2 xnorm of minimal energy blow up solutions is bounded from below, the L 2n n−2 xnorm is larger than the potential energy.
On blowup solutions to the 3D cubic nonlinear Schrödinger equation
 Appl. Math. Res. Express. AMRX
"... Abstract. For the 3d cubic nonlinear Schrödinger (NLS) equation, which has critical (scaling) norms L 3 and ˙ H 1/2, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finitetime blowup. For the rest of the paper, we focus on the study of ..."
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Cited by 20 (9 self)
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Abstract. For the 3d cubic nonlinear Schrödinger (NLS) equation, which has critical (scaling) norms L 3 and ˙ H 1/2, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finitetime blowup. For the rest of the paper, we focus on the study of finitetime radial blowup solutions, and prove a result on the concentration of the L 3 norm at the origin. Two disparate possibilities emerge, one which coincides with solutions typically observed in numerical experiments that consist of a specific bump profile with maximum at the origin and focus toward the origin at rate ∼ (T − t) 1/2, where T> 0 is the blowup time. For the other possibility, we propose the existence of “contracting sphere blowup solutions”, i.e. those that concentrate on a sphere of radius ∼ (T − t) 1/3, but focus towards this sphere at a faster rate ∼ (T − t) 2/3. These conjectured solutions are analyzed through heuristic arguments and shown (at this level of precision) to be consistent with all conservation laws of the equation. 1.