Abstract. 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.
We obtain global well-posedness, scattering, and globalspacetime bounds for energy-space solutions to the energy-critical nonlinear Schrödinger equation in Rt × R n x , n ≥ 5.
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 2x (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.
We obtain global well-posedness, scattering, uniform regularity, and global L 6 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 L 6 t,x -norm.
Abstract. We undertake a comprehensive study of the nonlinear Schrödinger equationwhere u(t, x) is a complex-valued function in spacetime Rt × R n x , λ 1 and λ 2 are nonzero real constants, and 0 < p 1 < p 2 ≤ 4 n−2. We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H 1 (R n ) and in the pseudoconformal space Σ := {f ∈ H 1 (R n ); xf ∈ L 2 (R n )}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L 2 x -critical, respectivelyḢ 1 x -critical NLS, that is, λ 1 , λ 2 > 0 and. The results at the endpoint p 1 = 4 n are conditional on a conjectured global existence and spacetime estimate for the L 2x -critical nonlinear Schrödinger equation.As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H 1 x for solutions to the nonlinear Schrödinger equation iut + ∆u = |u| p u,
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