Results 1  10
of
28
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.
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.
Global wellposedness and scattering for the mass critical nonlinear Schrödinger . . .
, 2013
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Global wellposedness of the MaxwellKleinGordon equation below the energy norm
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Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. arXiv:0909.0757
"... Abstract: We prove global wellposedness for the cubic, defocusing, nonlinear Schrödinger equation on R 2 with data u0 ∈ H s (R 2), s> 1/4. We accomplish this by improving the almost Morawetz estimates in [9]. 1 ..."
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Cited by 7 (6 self)
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Abstract: We prove global wellposedness for the cubic, defocusing, nonlinear Schrödinger equation on R 2 with data u0 ∈ H s (R 2), s> 1/4. We accomplish this by improving the almost Morawetz estimates in [9]. 1
Global rough solutions to the critical generalized KdV
, 908
"... We prove that the initial value problem (IVP) for the critical generalized KdV equation ut +uxxx+(u 5)x = 0 on the real line is globally wellposed in H s (R) in s> 3/5 with the appropriate smallness assumption on the initial data. 1 ..."
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Cited by 6 (1 self)
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We prove that the initial value problem (IVP) for the critical generalized KdV equation ut +uxxx+(u 5)x = 0 on the real line is globally wellposed in H s (R) in s> 3/5 with the appropriate smallness assumption on the initial data. 1
THE LOW REGULARITY GLOBAL SOLUTIONS FOR THE CRITICAL GENERALIZED KDV EQUATION
, 908
"... Abstract. In this paper, we establish the global wellposedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space Hs`R ´ with s> 1. Of course, we require that the mass is st ..."
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Cited by 6 (2 self)
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Abstract. In this paper, we establish the global wellposedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space Hs`R ´ with s> 1. Of course, we require that the mass is strictly less than 2 that of the ground state in the focusing case. This follows from “Imethod”, which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and improves the result in [20]. 1.
Global wellposedness for Schrödinger equation with derivative in H 2
 R), J. Diff. Eq
, 2011
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Almost Morawetz estimates and global wellposedness for the defocusing L 2critical nonlinear Schrödinger equation in higher dimensions
, 2009
"... Abstract: In this paper, we consider the global wellposedness of the defocusing, L2 critical nonlinear Schrödinger equation in dimensions n ≥ 3. Using the Imethod, we show the problem is globally wellposed in n = 3 when s> 2 n−2 5, and when n ≥ 4, for s> n. We combine energy increments for ..."
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Cited by 5 (5 self)
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Abstract: In this paper, we consider the global wellposedness of the defocusing, L2 critical nonlinear Schrödinger equation in dimensions n ≥ 3. Using the Imethod, we show the problem is globally wellposed in n = 3 when s> 2 n−2 5, and when n ≥ 4, for s> n. We combine energy increments for the Imethod, interaction Morawetz estimates, and almost Morawetz estimates to prove the result. 1