Title: Local Well-Posedness of the KdV Equation with Quasi-Periodic Initial Data
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 $f=f_1+f_2+\cdots+f_N$ w...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 $f=f_1+f_2+\cdots+f_N$ where $f_j$ is in the Sobolev space of order $s>-1/2N$ of $2\pi\alpha^{-1}_j$ periodic functions. Note that $f$ is not a periodic function when the ratio of periods $\alpha_i/\alpha_j$ 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.Read More