On the well-posedness of the generalized korteweg-de vries equation in scale-critical Lr-space

Satoshi Masaki; Jun Ichi Segata

doi:10.2140/apde.2016.9.699

On the well-posedness of the generalized korteweg-de vries equation in scale-critical L^r-space

Satoshi Masaki, Jun Ichi Segata

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

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 L^r = (f ε S' (ℝ): ||f||_L^r = ||f||_L^r' ≤ ∞). We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical L^r-space. A key ingredient is a Stein-Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for L^r-framework.

Original language	English
Pages (from-to)	699-725
Number of pages	27
Journal	Analysis and PDE
Volume	9
Issue number	3
DOIs	https://doi.org/10.2140/apde.2016.9.699
Publication status	Published - 2016
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Numerical Analysis
Applied Mathematics

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10.2140/apde.2016.9.699

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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 Lr = (f ε S' (ℝ): ||f||Lr = ||f||Lr' ≤ ∞). We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical Lr-space. A key ingredient is a Stein-Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for Lr-framework.",

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AB - 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 Lr = (f ε S' (ℝ): ||f||Lr = ||f||Lr' ≤ ∞). We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical Lr-space. A key ingredient is a Stein-Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for Lr-framework.

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On the well-posedness of the generalized korteweg-de vries equation in scale-critical L^r-space

Abstract

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