The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation

Zhaohui Huo; Boling Guo

doi:10.3934/dcds.2005.12.387

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2005, Volume 12, Issue 3: 387-402. Doi: 10.3934/dcds.2005.12.387

This issue Previous Article On the regularity of integrable conformal structures invariant under Anosov systems Next Article Stable sets, hyperbolicity and dimension

The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation

Zhaohui Huo^1, and
Boling Guo^2,

1.
Department of Mathematics, School of Sciences, Beijing University of Aeronautics and Astronautics, Beijing, 100083
2.
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088

Received: November 30, 2003

Revised: August 31, 2004

Published: March 2005

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Abstract

The well-posedness of the Cauchy problem for a generalized nonlinear dispersive equation is studied. Local well-posedness for data in $H^s(\mathbb R)(s>-\frac{1}{8})$ and the global result for data in $ L^{2}(\mathbb{R})$ are obtained if $l=2$. Moreover, for $l=3$, the problem is locally well-posed for data in $H^s(s>\frac{1}{4}).$ The main idea is to use the Fourier restriction norm method.

Keywords:

The generalized nonlinear dispersive equation,
the Fourier restriction norm,
[k;Z]-multiplier.

Mathematics Subject Classification: 35Q53, 35Q55, 35E15.

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