A bilinear Airy- estimate with application to gKdV-3

TL;DR

This paper establishes local well-posedness for the gKdV-3 equation using Fourier restriction norms and introduces a new bilinear estimate for the Airy equation, extending understanding of solution behavior in Sobolev spaces.

Contribution

It presents a novel bilinear Airy-estimate that improves the analysis of the gKdV-3 equation's well-posedness in Sobolev spaces.

Findings

Local well-posedness for s > -1/6 in Sobolev space H^s

Global well-posedness for real data in L^2 due to conservation laws

Introduction of a new bilinear estimate for the Airy equation

Abstract

The Fourier restriction norm method is used to show local wellposedness for the Cauchy Problem for the generalized KdV-equation of order three with data in the usual Sobolev space H^s, s > -1/6. For real valued data in L^2 global wellposedness follows by the conservation of the L^2-norm. The main new tool is a bilinear estimate for solutions of the Airy- equation.