Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension d=4

2019 ◽  
Vol 52 (1) ◽  
pp. 139-180 ◽  
Author(s):  
Benjamin DODSON
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Well Posedness ◽  
Cubic Schrödinger Equation

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide ℝ2 × 𝕋2

2019 ◽  
Vol 16 (01) ◽  
pp. 73-129 ◽  
Author(s):  
Zehua Zhao
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Data ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Resonant System ◽  
Induction Method ◽  
Profile Decomposition ◽  
Cubic Schrödinger Equation ◽  
Cubic Nonlinear Schrödinger Equation

We consider the problem of large data scattering for the defocusing cubic nonlinear Schrödinger equation on [Formula: see text]. This equation is critical both at the level of energy and mass. The key ingredients are global-in-time Stricharz estimate, resonant system approximation, profile decomposition and energy induction method. Assuming the large data scattering for the 2d cubic resonant system, we prove the large data scattering for this problem. This problem is the cubic analogue of a problem studied by Hani and Pausader.

scholarly journals ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 549-575 ◽  
Author(s):  
J. COLLIANDER ◽  
S. IBRAHIM ◽  
M. MAJDOUB ◽  
N. MASMOUDI
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Supercritical Case ◽  
Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

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