Abstract
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 $s \geq -3/2$ by a variant of the Fourier restriction norm method introduced by Bourgain. Moreover, these local solutions can be extended globally in time for $s \geq -1$ by the I-method. On the other hand, we prove ill-posedness for $s < -3/2$ in some sense. This is a sharp contrast to the results in the case of $\mathbb{R}$, where the critical exponent is equal to $-2$.
Citation
Takamori Kato. "Low regularity well-posedness for the periodic Kawahara equation." Differential Integral Equations 25 (11/12) 1011 - 1036, November/December 2012. https://doi.org/10.57262/die/1356012249
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