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On the well-posedness problem for the derivative nonlinear Schrödinger equation

Rowan Killip, Maria Ntekoume and Monica Vişan

Vol. 16 (2023), No. 5, 1245–1270
Abstract

We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L2-critical with respect to scaling. We first discuss whether ensembles of orbits with L2-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction M(q) =|q|2 < 4π. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well posed for initial data in H16 under the same restriction on M. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.

Keywords
derivative nonlinear Schrödinger equation, well-posedness
Mathematical Subject Classification
Primary: 35Q55
Milestones
Received: 29 January 2021
Revised: 28 June 2021
Accepted: 6 October 2021
Published: 12 August 2023
Authors
Rowan Killip
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States
Maria Ntekoume
Department of Mathematics
Rice University
Houston, TX
United States
Monica Vişan
Math Sciences
University of California, Los Angeles
Los Angeles, CA
United States

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