Vol. 9, No. 3, 2016

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On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical $\hat{L}^r$-space

Satoshi Masaki and Jun-ichi Segata

Vol. 9 (2016), No. 3, 699–725
Abstract

The purpose of this paper is to study local and global well-posedness of the initial value problem for the generalized Korteweg–de Vries (gKdV) equation in L̂r = {f S() : fL̂r = f̂Lr < }. We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical L̂r-space. A key ingredient is a Stein–Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for L̂r-framework.

Keywords
generalized Korteweg–de Vries equation, scattering problem
Mathematical Subject Classification 2010
Primary: 35Q53, 35B40
Secondary: 35B30
Milestones
Received: 28 July 2015
Revised: 17 November 2015
Accepted: 30 January 2016
Published: 17 June 2016
Authors
Satoshi Masaki
Laboratory of Mathematics
Institute of Engineering
Hiroshima University
Higashihiroshima, Hiroshima 739-8527
Japan
Jun-ichi Segata
Mathematical Institute
Tohoku University
6-3, Aoba, Aramaki, Aoba-ku
Sendai 980-8578
Japan