Global Solutions of Nonlinear Schrodinger Equations

By Y. Ye. Journal of Partial Differential Equations, 2005

344 citations recorded by the source corpus.

In this paper we study the existence of global solutions to the Cauchy problem of nonlinear SchrSdinger equation by establishing time weight function spaces and using the contraction mapping principle.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f9fc9ad339fa4989fc6b5d518492ae5a32b2c968

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Citations in the corpus

INVARIANT WEIGHTED WIENER MEASURES AND ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE PERIODIC DERIVATIVE NLS. (2010, background): “In those cases, it seems natural to return to one of Bourgain’s early approaches for periodic dispersive equations (NLS, KdV, mKdV, Zakharov system) [3, 4, 5, 7, 8, 9] where global in time existence was studied in the almost sure sense via the existence and invariance of the associated Gibbs measure”
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 (2003, background): “Also, the bounds we obtain on 3 In [2, 3], it is also shown that the difference between the linear and nonlinear evolutions from rough data has finite energy.”
Quantization of Pseudo-differential Operators on the Torus (2008, methodology): “We also note that this approach to the representation of solutions is related to the work of Bourgain [5, 6, 7] on Strichartz estimates for solutions to Schrödinger equations on the torus T.”
On universality of blow-up profile for L2 critical nonlinear Schrödinger equation (2004, background): “See also [4], [7] for the periodic case and global well posedness results.”
Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems (2005, unknown): “Keywords: Feynman diagrams, BBGKY hierarchy, dispersive estimates, propagation of chaos.”
The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, background): “5], which builds on earlier versions in [6, 14].”
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation (2003, background): “See also [3], [2] for the periodic case and global well posedness results.”
Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schr\"odinger Equations (2015, background): “Some highlights of the developments are [16, 18, 27, 29, 39, 40, 53, 54, 69, 72], for example.”
Radial symmetry and monotonicity for an integral equation (2008, background): “(3) Such a kind of equation also arises from the ground states of the Schrödinger equation [2].”
Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “5], which builds on earlier versions in [8, 20].”
Regularity properties of the cubic nonlinear Schr\"odinger equation on the half line (2015, methodology): “We note that smoothing results of this type were first obtained by Bourgain, see [5], for the cubic NLS on R(2).”
Growth of Sobolev Norms of Solutions of Linear Schrödinger Equations on Some Compact Manifolds (2009, methodology): “The proof of Theorem 1.1 given by Bourgain in [4] (see also appendix 1 of [5] for the case d = 1) relies on reduction to a time periodic problem and then uses space-time Fourier series decompositions.”
Dynamic of Threshold Solutions for Energy-Critical NlS (2007, background): “From [ B1 ,2] we know that a solution such that k uk S(R) < ∞ scatters in u H 1 at ±∞. Cases (a) and (b) of Theorem 2 show:”
Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, methodology): “The latter proof is a slight modification of a proof from [15].”
On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations (2006, background): “Global well-posedness in Ḣ 1 x (R 3) for the energy-critical NLS in the case of large finiteenergy, radially-symmetric initial data was first obtained by Bourgain [2,3] and subsequently by Grillakis [11].”