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48
Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear 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 well-posedness and scattering for solutions to the mass-critical 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 well-posedness and scattering for solutions to the mass-critical 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 well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4
, 2006
"... We obtain global well-posedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequency-localized inte ..."
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Cited by 70 (15 self)
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We obtain global well-posedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the L6 t,x-norm
Minimal-mass blowup solutions of the mass-critical NLS
, 2006
"... We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + ∆u = µ|u | 4/d u to blow up. If m0 is finite, we show that there exists a minimal-mass 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 mass-critical nonlinear Schrödinger (NLS) equation iut + ∆u = µ|u | 4/d u to blow up. If m0 is finite, we show that there exists a minimal-mass 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 mass-critical NLS and by Kenig and Merle, [18], in the energy-critical 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 power-type nonlinearities
, 2005
"... We undertake a comprehensive study of the nonlinear Schrödinger equation iut + ∆u = λ1|u | p1 u + λ2|u | p2 u, where u(t, x) is a complex-valued 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 = λ1|u | p1 u + λ2|u | p2 u, where u(t, x) is a complex-valued 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 well-posedness, 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 x-critical, respectively ˙ H1 x-critical NLS, that is, λ1, λ2> 0 and p1 = 4 n, p2 = 4
THE FOCUSING ENERGY-CRITICAL NONLINEAR SCHRÖDINGER EQUATION IN DIMENSIONS FIVE AND HIGHER
"... Abstract. We consider the focusing energy-critical nonlinear Schrödinger equation iut + ∆u = 4 −|u | d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan 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 energy-critical nonlinear Schrödinger equation iut + ∆u = 4 −|u | d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan 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 mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
"... Abstract. We establish global well-posedness and scattering for solutions to the mass-critical 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 well-posedness and scattering for solutions to the mass-critical 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 well-posedness and scattering for the mass-critical nonlinear Schrödinger equations for radial data in high dimensions
, 2006
"... We establish global well-posedness and scattering for solutions to the defocusing mass-critical (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 well-posedness and scattering for solutions to the defocusing mass-critical (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 frequency-localized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to [9], [23], [36]) in order to conclude the argument.
Global well-posedness and scattering for the defocusing L²-critical 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.
