A Refined Global Well-Posedness Result for Schrödinger Equations with Derivative

Abstract

In this paper we prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term is globally well-posed in Hs for $s > \frac12$ for data small in L2 . To understand the strength of this result one should recall that for $s < \frac12$ the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the "I-method" used by the same authors to obtain global well-posedness for $s >\frac23$. The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in Hs for $s>\frac12$.

Locations

• SIAM Journal on Mathematical Analysis - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• SIAM Journal on Mathematical Analysis - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF