On well-posedness of generalized Korteweg-de Vries equation in scale critical ^L^r space

TL;DR

This paper investigates the well-posedness of the generalized Korteweg-de Vries (gKdV) equation in scale-critical L^r spaces, establishing local and global results, and introducing a Stein-Tomas type inequality for the Airy equation.

Contribution

It introduces a Stein-Tomas type inequality for the Airy equation, enabling analysis of gKdV in scale-critical L^r spaces, including large data local and small data global well-posedness.

Findings

Large data local well-posedness in L^r

Small data global well-posedness in L^r

Small data scattering in L^r

Abstract

The purpose of this paper is to study local and global well-posedness of initial value problem for generalized Korteweg-de Vries (gKdV) equation in ^L^r. We show (large data) local well-posedness, small data global well-posedness, and small data scattering for gKdV equation in the scale critical ^L^r space. A key ingredient is a Stein-Tomas type inequality for the Airy equation, which generalizes usual Strichartz estimates for ^L^r-framework.