Results 1 - 10 of 13

Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4

by E. Ryckman, M. Visan , 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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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

The nonlinear Schrödinger equation with combined power-type nonlinearities

by Terence Tao, Monica Visan, Xiaoyi Zhang , 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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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 &lt; p1 &lt; 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&gt; 0 and p1 = 4 n, p2 = 4

Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data

by Terence Tao - J. MATH , 2004
"... In any dimension n ≥ 3, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear 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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In any dimension n ≥ 3, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear 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 tower-type bounds of Bourgain. In higher dimensions n ≥ 6 some new technical difficulties arise because of the very low power of the non-linearity.
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Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equations for radial data in high dimensions

by Terence Tao, Monica Visan, Xiaoyi Zhang , 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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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.

A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds

by Andrew Hassell, Terence Tao, Jared Wunsch - Comm. Partial Differential Equations
"... Abstract. We obtain the Strichartz inequality ..."
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Abstract. We obtain the Strichartz inequality
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Pair excitations and the mean field approximation of interacting Bosons

by M Grillakis , M Machedon , 1208
"... Abstract. In our previous work ..."
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Abstract. In our previous work
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Global well-posedness and scattering for a class of nonlinear Schrödinger equations below the energy space

by Monica Visan, Xiaoyi Zhang , 2006
"... We prove global well-posedness and scattering for the nonlinear Schrödinger equation with power-type 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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We prove global well-posedness and scattering for the nonlinear Schrödinger equation with power-type nonlinearity iut + ∆u = |u | p 4 4 u, &lt; p &lt; n n−2, u(0, x) = u0(x) ∈ Hs (Rn), n ≥ 3, below the energy space, i.e., for s &lt; 1. In [14], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the H s x-norm of the solution, and hence global well-posedness 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 &lt; s0(n, p) &lt; 1.

GLOBAL BEHAVIOUR OF NONLINEAR DISPERSIVE AND WAVE EQUATIONS

by Terence Tao , 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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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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Energy-critical NLS with . . .

by Rowan Killip, Monica Visan, Xiaoyi Zhang , 2006
"... We consider the defocusing ˙ H1-critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1 2 |x|2. We show global well-posedness 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 ˙ H1-critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1 2 |x|2. We show global well-posedness 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.

unknown title

by unknown authors
"... Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data ..."
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Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data
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