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  1. Calculus of Variations and Partial Differential Equations
  2. Calculus of Variations and Partial Differential Equations : Volume 44
  3. Calculus of Variations and Partial Differential Equations : Volume 44, Issue 1-2, May 2012
  4. Global dynamics above the ground state energy for the cubic NLS equation in 3D
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Calculus of Variations and Partial Differential Equations : Volume 56
Calculus of Variations and Partial Differential Equations : Volume 55
Calculus of Variations and Partial Differential Equations : Volume 54
Calculus of Variations and Partial Differential Equations : Volume 53
Calculus of Variations and Partial Differential Equations : Volume 52
Calculus of Variations and Partial Differential Equations : Volume 51
Calculus of Variations and Partial Differential Equations : Volume 50
Calculus of Variations and Partial Differential Equations : Volume 49
Calculus of Variations and Partial Differential Equations : Volume 48
Calculus of Variations and Partial Differential Equations : Volume 47
Calculus of Variations and Partial Differential Equations : Volume 46
Calculus of Variations and Partial Differential Equations : Volume 45
Calculus of Variations and Partial Differential Equations : Volume 44
Calculus of Variations and Partial Differential Equations : Volume 44, Issue 3-4, July 2012
Calculus of Variations and Partial Differential Equations : Volume 44, Issue 1-2, May 2012
Global dynamics above the ground state energy for the cubic NLS equation in 3D
A proof by calibration of an isoperimetric inequality in the Heisenberg group $${\mathbb{H}^n}$$
A nonhomogeneous boundary value problem in mass transfer theory
Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape
New regularity theorems for non-autonomous variational integrals with (p, q)-growth
Surface diffusion flow near spheres
Strain-gradient theory of hydroelastic travelling waves and young measures of their singular limits
Lagrangian mean curvature flow for entire Lipschitz graphs
Singular limit of an energy minimizer arising from dewetting thin film model with van der Waal, born repulsion and surface tension forces
Poincaré–Sobolev equations in the hyperbolic space
Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model
Calculus of Variations and Partial Differential Equations : Volume 43
Calculus of Variations and Partial Differential Equations : Volume 42
Calculus of Variations and Partial Differential Equations : Volume 41
Calculus of Variations and Partial Differential Equations : Volume 40
Calculus of Variations and Partial Differential Equations : Volume 39
Calculus of Variations and Partial Differential Equations : Volume 38
Calculus of Variations and Partial Differential Equations : Volume 37
Calculus of Variations and Partial Differential Equations : Volume 36
Calculus of Variations and Partial Differential Equations : Volume 35
Calculus of Variations and Partial Differential Equations : Volume 34
Calculus of Variations and Partial Differential Equations : Volume 33
Calculus of Variations and Partial Differential Equations : Volume 32
Calculus of Variations and Partial Differential Equations : Volume 31
Calculus of Variations and Partial Differential Equations : Volume 30
Calculus of Variations and Partial Differential Equations : Volume 29
Calculus of Variations and Partial Differential Equations : Volume 28
Calculus of Variations and Partial Differential Equations : Volume 27
Calculus of Variations and Partial Differential Equations : Volume 26
Calculus of Variations and Partial Differential Equations : Volume 25
Calculus of Variations and Partial Differential Equations : Volume 24
Calculus of Variations and Partial Differential Equations : Volume 23
Calculus of Variations and Partial Differential Equations : Volume 22
Calculus of Variations and Partial Differential Equations : Volume 21
Calculus of Variations and Partial Differential Equations : Volume 20
Calculus of Variations and Partial Differential Equations : Volume 19
Calculus of Variations and Partial Differential Equations : Volume 18
Calculus of Variations and Partial Differential Equations : Volume 17
Calculus of Variations and Partial Differential Equations : Volume 16
Calculus of Variations and Partial Differential Equations : Volume 15
Calculus of Variations and Partial Differential Equations : Volume 14
Calculus of Variations and Partial Differential Equations : Volume 13
Calculus of Variations and Partial Differential Equations : Volume 12
Calculus of Variations and Partial Differential Equations : Volume 11
Calculus of Variations and Partial Differential Equations : Volume 10
Calculus of Variations and Partial Differential Equations : Volume 9
Calculus of Variations and Partial Differential Equations : Volume 8
Calculus of Variations and Partial Differential Equations : Volume 7
Calculus of Variations and Partial Differential Equations : Volume 6
Calculus of Variations and Partial Differential Equations : Volume 5

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Global dynamics above the ground state energy for the cubic NLS equation in 3D

Content Provider SpringerLink
Author Schlag, W. Nakanishi, K.
Copyright Year 2011
Abstract We extend the result in Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) on the nonlinear Klein–Gordon equation to the nonlinear Schrödinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which contains the ground states and separates the phase space locally into two connected regions exhibiting blow-up and scattering to 0, respectively. The special solutions found by Duyckaerts and Roudenko (Rev Mater Iberoam 26(1):1–56, 2010), following the seminal work on threshold solutions by Duyckaerts and Merle (Funct Anal 18(6):1787–1840, 2009), appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011), the proof combines the hyperbolic dynamics near the ground states with the variational structure away from them. The main technical ingredient in the proof is a “one-pass” theorem which precludes “almost homoclinic orbits”, i.e., those solutions starting in, then moving away from, and finally returning to, a small neighborhood of the ground states. The main new difficulty compared with the Klein–Gordon case is the lack of finite propagation speed. We need the radial Sobolev inequality for the error estimate in the virial argument. Another major difference between Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) and this paper is the need to control two modulation parameters.
Ending Page 45
Page Count 45
Starting Page 1
File Format PDF
ISSN 09442669
e-ISSN 14320835
Journal Calculus of Variations and Partial Differential Equations
Issue Number 1-2
Volume Number 44
Language English
Publisher Springer-Verlag
Publisher Date 2011-04-21
Publisher Place Berlin, Heidelberg
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Estimation of eigenvalues, upper and lower bounds Soliton theory, asymptotic behavior of solutions Calculus of Variations and Optimal Control; Optimization Systems Theory, Control Analysis Theoretical, Mathematical and Computational Physics Invariant manifold theory NLS-like equations (nonlinear Schrödinger) Stability problems
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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