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

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

2011-09-30

arxiv.org/abs/1110.0046v1

Mathematics

Analysis of PDEs

20 pages

Scientific paper

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 quasi periodic 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 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.

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