Local well-posedness of the KdV equation with quasi periodic initial data

TL;DR

This paper establishes local well-posedness for the KdV equation with quasi-periodic initial data in a specific function space, extending the understanding of solutions beyond purely periodic cases and highlighting ill-posedness issues.

Contribution

It proves local well-posedness for the KdV equation with quasi-periodic initial data and introduces a new function space framework for such analysis.

Findings

Proves local well-posedness in a quasi-periodic function space.

Demonstrates ill-posedness via the flow map not being C^2.

Utilizes Fourier restriction norm method for the proof.

Abstract

We prove the local well-posedness for the Cauchy problem of the Korteweg-de Vries equation in a quasi periodic function space. The function space contains functions such that f=f_1+f_2+...+f_N where f_j is in the Sobolev space of order s>-1/2N of a_j periodic functions. Note that f is not a periodic function when the ratio of periods a_i/a_j is irrational. The main tool of the proof is the Fourier restriction norm method introduced by Bourgain. We also prove an ill-posedness result in the sense that the flow map (if it exists) is not C^2, which is related to the Diophantine problem.