Global well-posedness for Schrödinger equation with derivative in H 1 2 ( R )

By Changxing Miao, Yifei Wu, Guixiang Xu. 2009

59 citations recorded by the source corpus.

Abstract In this paper, we consider the Cauchy problem of the cubic nonlinear Schrodinger equation with derivative in H s ( R ) . This equation was known to be the local well-posedness for s ⩾ 1 2 (Takaoka, 1999 [27] ), ill-posedness for s 1 2 (Biagioni and Linares, 2001 [1] , etc.) and global well-posedness for s > 1 2 (I-team, 2002 [10] ). In this paper, we show that it is global well-posedness in the endpoint space H 1 2 ( R ) , which remained open previously. The main approach is the third generation I-method combined with a new resonant decomposition technique. The resonant decomposition is applied to control the singularity coming from the resonant interaction.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/51422c04c828421bc639b4ec6228d927bc88cbcc

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Global well-posedness on the derivative nonlinear Schrödinger equation (2014, background): “Results on the global wellposedness when the initial data of regularity below the energy space, see [4, 5, 10, 16] for examples.”
INVARIANT WEIGHTED WIENER MEASURES AND ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE PERIODIC DERIVATIVE NLS. (2010, background): “Local well-posedness is known for σ ≥ 1/2 for the nonperiodic [40] and periodic [26] cases while global well-posedness is known for σ ≥ 1/2 for the nonperiodic case (σ > 1/2 in [16] and σ ≥ 1/2 in [31]) and for σ > 1/2 in the periodic case [47].”