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

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

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Kotaro Tsugawa, Nagoya University/University of Toronto
Fine Hall 314

We prove the local well-posedness for the Cauchy problem of Korteweg-de Vries equation in an almost periodic function space. The function space contains functions satisfying $f=f_1+f_2+...+f_N$ where $f_j$ is in the Sobolev space of order $s>?1/2N$ of $a_j$ periodic functions. Note that f is not periodic 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 C2, which is related to the Diophantine problem.