Low regularity well-posedness for the periodic Kawahara equation. (English) Zbl 1274.35359

Kato, Takamori

Differ. Integral Equ. 25, No. 11-12, 1011-1036 (2012).

Summary: 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 \(-2\).

MSC:

35Q55

NLS equations (nonlinear Schrödinger equations)

Keywords:

well-posedness; Kawahara equation; critical exponent