Low regularity data for the periodic Kawahara equation

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 $s\ge -3/2$ by a variant of the Fourier restriction norm method. On the other hand, we show the ill-posedness for $s<-3/2$ in weak sense. Moreover, the local solutions can be extended globally in time for $s\ge -1$ by the I-method. This is a shape contrast to the results in the non-periodic setting, where the critical exponent is equal to -2.