Low regularity well-posedness for the periodic Kawahara equation

In this paper, we consider the well-posedness for the Cauchy problem of the Kawahara equation with low regularity data in the periodic case. We obtain the local well-posedness for s ≥ −3/2 by a variant of Fourier restriction norm method introduced by Bourgain. Moreover, these local solutions can be extended globally in time for s ≥ −1 by the I-method. On the other hand, we prove illposedness for s < −3/2 in some sense. This is a shape contrast to the results in the case of R, where the critical exponent is equal to −2.

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