Results 1  10
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88
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 EXISTENCE AND SCATTERING FOR ROUGH SOLUTIONS OF A NONLINEAR SCHRÖDINGER EQUATION ON R³
, 2003
"... We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs (R3) for s> 4. The main new estimate in the argument is a Morawetztype inequality for the solution φ. 5 This estimate bounds ‖φ(x, t)‖L4 x,t (R3×R), whereas the wellknown Morawetztype estima ..."
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Cited by 69 (16 self)
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We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs (R3) for s> 4. The main new estimate in the argument is a Morawetztype inequality for the solution φ. 5 This estimate bounds ‖φ(x, t)‖L4 x,t (R3×R), whereas the wellknown Morawetztype estimate of LinStrauss controls
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.
A refined global wellposedness result for Schrödinger equations with derivative
 SIAM J. Math. Anal
, 2002
"... Abstract. In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 1 2 for data small in L2. To understand the strength of this result one should recall that for s < 1 the Cauchy problem is illposed, in the 2 sense tha ..."
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Cited by 59 (18 self)
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Abstract. In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 1 2 for data small in L2. To understand the strength of this result one should recall that for s < 1 the Cauchy problem is illposed, in the 2 sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the “Imethod ” used by the same authors. The same argument can be used to prove that any to obtain global wellposedness for s> 2 3 quintic nonlinear defocusing Schrödinger equation on the line is globally wellposed for large data in H s, for s> 1 2. 1.
Multilinear estimates for periodic KdV equations, and applications
"... Abstract We prove an endpoint multilinear estimate for the X s;b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local wellposedness results for periodic generalized KdV equations, as well as some global wellposedness results below the energy norm. r ..."
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Cited by 56 (14 self)
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Abstract We prove an endpoint multilinear estimate for the X s;b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local wellposedness results for periodic generalized KdV equations, as well as some global wellposedness results below the energy norm. r
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 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.
Resonant decompositions and the Imethod for cubic nonlinear Schrödinger
 on R 2 . Disc. Cont. Dynam. Systems A
"... Abstract. The initial value problem for the cubic defocusing nonlinear Schrödinger equation i∂tu + ∆u = u  2 u on the plane is shown to be globally wellposed for initial data in H s (R 2) provided s> 1/2. The proof relies upon an almost conserved quantity constructed using multilinear correcti ..."
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Cited by 28 (4 self)
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Abstract. The initial value problem for the cubic defocusing nonlinear Schrödinger equation i∂tu + ∆u = u  2 u on the plane is shown to be globally wellposed for initial data in H s (R 2) provided s> 1/2. The proof relies upon an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms. The resonant interactions are significant due to the multidimensional setting of the problem and some orthogonality issues which arise. 1.
LOW REGULARITY GLOBAL WELLPOSEDNESS FOR THE ZAKHAROV AND KLEINGORDONSCHRÖDINGER SYSTEMS
"... Abstract. We prove lowregularity global wellposedness for the 1d Zakharov system and 3d KleinGordonSchrödinger system, which are systems in two variables u: R d x × Rt → C and n: R d x × Rt → R. The Zakharov system is known to be locally wellposed in (u, n) ∈ L2 × H−1/2 and the KleinGordonSc ..."
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Cited by 20 (2 self)
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Abstract. We prove lowregularity global wellposedness for the 1d Zakharov system and 3d KleinGordonSchrödinger system, which are systems in two variables u: R d x × Rt → C and n: R d x × Rt → R. The Zakharov system is known to be locally wellposed in (u, n) ∈ L2 × H−1/2 and the KleinGordonSchrödinger system is known to be locally wellposed in (u, n) ∈ L2 × L2. Here, we show that the Zakharov and KleinGordonSchrödinger systems are globally wellposed in these spaces, respectively, by using an available conservation law for the L 2 norm of u and controlling the growth of n via the estimates in the local theory.