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13
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
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
Global wellposedness and scattering for the higherdimensional energycritical nonlinear Schrödinger equation for radial data
 J. MATH
, 2004
"... In any dimension n ≥ 3, we show that spherically symmetric bounded energy solutions of the defocusing energycritical nonlinear Schrödinger equation iut+∆u = u  4 n−2 u in R×Rn exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain [1], ..."
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Cited by 48 (8 self)
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In any dimension n ≥ 3, we show that spherically symmetric bounded energy solutions of the defocusing energycritical nonlinear Schrödinger equation iut+∆u = u  4 n−2 u in R×Rn exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain [1], [2] and Grillakis [11]. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the towertype bounds of Bourgain. In higher dimensions n ≥ 6 some new technical difficulties arise because of the very low power of the nonlinearity.
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.
A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds
 Comm. Partial Differential Equations
"... Abstract. We obtain the Strichartz inequality ..."
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Pair excitations and the mean field approximation of interacting Bosons
, 1208
"... Abstract. In our previous work ..."
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Global wellposedness and scattering for a class of nonlinear Schrödinger equations below the energy space
, 2006
"... We prove global wellposedness and scattering for the nonlinear Schrödinger equation with powertype nonlinearity iut + ∆u = u  p 4 4 u, < p < n n−2, u(0, x) = u0(x) ∈ Hs (Rn), n ≥ 3, below the energy space, i.e., for s < 1. In [14], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, ..."
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Cited by 13 (0 self)
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We prove global wellposedness and scattering for the nonlinear Schrödinger equation with powertype nonlinearity iut + ∆u = u  p 4 4 u, < p < n n−2, u(0, x) = u0(x) ∈ Hs (Rn), n ≥ 3, below the energy space, i.e., for s < 1. In [14], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the H s xnorm of the solution, and hence global wellposedness for initial data in H s x, provided s is sufficiently close to 1. However, their bounds are insufficient to yield scattering. In this paper, we use the a priori interaction Morawetz inequality to show that scattering holds in H s (R n) whenever s is larger than some value 0 < s0(n, p) < 1.
GLOBAL BEHAVIOUR OF NONLINEAR DISPERSIVE AND WAVE EQUATIONS
, 2006
"... Abstract. We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear ..."
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Cited by 9 (3 self)
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Abstract. We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear
Energycritical NLS with . . .
, 2006
"... We consider the defocusing ˙ H1critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1 2 x2. We show global wellposedness for radial initial data obeying ∇u0(x), xu0(x) ∈ L2. In view of the potential V, this is the natural energy space. In the ..."
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We consider the defocusing ˙ H1critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1 2 x2. We show global wellposedness for radial initial data obeying ∇u0(x), xu0(x) ∈ L2. In view of the potential V, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential.
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"... Global wellposedness and scattering for the higherdimensional energycritical nonlinear Schrödinger equation for radial data ..."
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Global wellposedness and scattering for the higherdimensional energycritical nonlinear Schrödinger equation for radial data