Well-posedness and smoothing effect for generalized nonlinear Schrödinger equations

Bienaimé, Pierre-Yves; Boulkhemair, Abdesslam

doi:10.2140/apde.2018.11.1241

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Well-posedness and smoothing effect for generalized nonlinear Schrödinger equations

Pierre-Yves Bienaimé and Abdesslam Boulkhemair

Vol. 11 (2018), No. 5, 1241–1284

DOI: 10.2140/apde.2018.11.1241

Abstract

We improve the result obtained by one of the authors, Bienaimé (2014), and establish the well-posedness of the Cauchy problem for some nonlinear equations of Schrödinger type in the usual Sobolev space $H^{s} (ℝ^{n})$ for $s > \frac{n}{2} + 2$ instead of $s > \frac{n}{2} + 3$ . We also improve the smoothing effect of the solution and obtain the optimal exponent.

Keywords

Cauchy problem, well-posedness, smoothing effect, nonlinear equation, Schrödinger, paradifferential, pseudodifferential, operator, paralinearization

Mathematical Subject Classification 2010

Primary: 47G20, 47G30

Milestones

Received: 20 June 2017

Revised: 26 November 2017

Accepted: 2 January 2018

Published: 11 April 2018

Authors

Pierre-Yves Bienaimé
Laboratoire de Mathématiques Jean Leray CNRS UMR6629, Université de Nantes Nantes France

Abdesslam Boulkhemair
Laboratoire de Mathématiques Jean Leray CNRS UMR6629, Université de Nantes Nantes France

Vol. 11, No. 5, 2018