Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrödinger Equation in $H^s(\mathbb{R}^n)$

Global Well - Posedness and Scattering for the Energy - Critical Nonlinear Schrödinger Equation In

We obtain global well-posedness, scattering, and global L10 t,x spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation in R1+3, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial case. The method is similar in spirit to the...

6 Global Well - Posedness and Scattering for the Defocusing Energy - Critical Nonlinear Schrödinger Equation in R

We obtain global well-posedness, scattering, uniform regularity, and global L6t,x spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schrödinger equation in R × R. 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 o...

Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data

Abstract: For n > 3, we study the Cauchy problem for the fourth order nonlinear Schrödinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces Bs 2,1(R n) are obtained. In one spatial dimension, we get the global well-posedness result with small data in the critical homogeneous Besov spaces Ḃs 2,1. As a by-pr...

Global Well - Posedness and Scattering for the Defocusing , L 2 - Critical Nonlinear Schrödinger

Global Well-posedness and Scattering for the Focusing Nonlinear Schrödinger Equation in the Nonradial Case

The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: i∂tu = −∆u− |u| 4 N−2 u, u(x, 0) = u0 ∈ H(R ), N ≥ 3. Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem pro...