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48
Global wellposedness, scattering, and blowup for the energycritical, focusing, nonlinear Schrödinger equation in the radial case
, 2006
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The cubic nonlinear Schrödinger equation in two dimensions with radial data
, 2008
"... We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut + ∆u = ±u  2 u for large spherically symmetric L 2 x(R 2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state ..."
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Cited by 90 (14 self)
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We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut + ∆u = ±u  2 u for large spherically symmetric L 2 x(R 2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.
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
Minimalmass blowup solutions of the masscritical NLS
, 2006
"... We consider the minimal mass m0 required for solutions to the masscritical nonlinear Schrödinger (NLS) equation iut + ∆u = µu  4/d u to blow up. If m0 is finite, we show that there exists a minimalmass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, wh ..."
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Cited by 63 (20 self)
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We consider the minimal mass m0 required for solutions to the masscritical nonlinear Schrödinger (NLS) equation iut + ∆u = µu  4/d u to blow up. If m0 is finite, we show that there exists a minimalmass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in L 2 x (Rd) is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [17], in dimensions 1, 2 and Begout and Vargas, [2], in dimensions d ≥ 3 for the masscritical NLS and by Kenig and Merle, [18], in the energycritical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in L 2 x (Rd) for the defocusing NLS in three and higher dimensions with spherically symmetric data.
Nonlinear Schrödinger equations at critical regularity
 CLAY LECTURE NOTES
, 2009
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The nonlinear Schrödinger equation with combined powertype nonlinearities
, 2005
"... We undertake a comprehensive study of the nonlinear Schrödinger equation iut + ∆u = λ1u  p1 u + λ2u  p2 u, where u(t, x) is a complexvalued function in spacetime Rt × Rn x, λ1 and λ2 are nonzero real constants, and 0 < p1 < p2 ≤ 4. We address questions n−2 related to local and global we ..."
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Cited by 54 (13 self)
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We undertake a comprehensive study of the nonlinear Schrödinger equation iut + ∆u = λ1u  p1 u + λ2u  p2 u, where u(t, x) is a complexvalued function in spacetime Rt × Rn x, λ1 and λ2 are nonzero real constants, and 0 < p1 < p2 ≤ 4. We address questions n−2 related to local and global wellposedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H1 (Rn) and in the pseudoconformal space Σ: = {f ∈ H1 (Rn); xf ∈ L2 (Rn)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2 xcritical, respectively ˙ H1 xcritical NLS, that is, λ1, λ2> 0 and p1 = 4 n, p2 = 4
THE FOCUSING ENERGYCRITICAL NONLINEAR SCHRÖDINGER EQUATION IN DIMENSIONS FIVE AND HIGHER
"... Abstract. We consider the focusing energycritical nonlinear Schrödinger equation iut + ∆u = 4 −u  d−2 u in dimensions d ≥ 5. We prove that if a maximallifespan solution u: I ×Rd → C obeys supt∈I ‖∇u(t)‖2 < ‖∇W‖2, then it is global and scatters both forward and backward in time. Here W denotes ..."
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Cited by 54 (8 self)
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Abstract. We consider the focusing energycritical nonlinear Schrödinger equation iut + ∆u = 4 −u  d−2 u in dimensions d ≥ 5. We prove that if a maximallifespan solution u: I ×Rd → C obeys supt∈I ‖∇u(t)‖2 < ‖∇W‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle for spherically symmetric initial data and dimensions d =3,4,5.
The masscritical nonlinear Schrödinger equation with radial data in dimensions three and higher
"... Abstract. We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut+∆u = ±u  4/d u for large spherically symmetric L 2 x (Rd) initial data in dimensions d ≥ 3. In the focusing case we require that the mass is strictly less than that of ..."
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Cited by 52 (10 self)
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Abstract. We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut+∆u = ±u  4/d u for large spherically symmetric L 2 x (Rd) initial data in dimensions d ≥ 3. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. 1.
Global wellposedness and scattering for the masscritical nonlinear Schrödinger equations for radial data in high dimensions
, 2006
"... We establish global wellposedness and scattering for solutions to the defocusing masscritical (pseudoconformal) nonlinear Schrödinger equation iut + ∆u = u  4/n u for large spherically symmetric L 2 x(R n) initial data in dimensions n ≥ 3. After using the reductions in [32] to reduce to elimina ..."
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Cited by 39 (20 self)
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We establish global wellposedness and scattering for solutions to the defocusing masscritical (pseudoconformal) nonlinear Schrödinger equation iut + ∆u = u  4/n u for large spherically symmetric L 2 x(R n) initial data in dimensions n ≥ 3. After using the reductions in [32] to reduce to eliminating blowup solutions which are almost periodic modulo scaling, we obtain a frequencylocalized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to [9], [23], [36]) in order to conclude the argument.
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.