Differential and Integral Equations

Low regularity well-posedness for the periodic Kawahara equation

Takamori Kato

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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 $-2$.

Article information

Source
Differential Integral Equations, Volume 25, Number 11/12 (2012), 1011-1036.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012249

Mathematical Reviews number (MathSciNet)
MR3013402

Zentralblatt MATH identifier
1274.35359

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

Kato, Takamori. Low regularity well-posedness for the periodic Kawahara equation. Differential Integral Equations 25 (2012), no. 11/12, 1011--1036. https://projecteuclid.org/euclid.die/1356012249


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