Resonant decomposition and the $I$-method for the two-dimensional Zakharov system

Nobu Kishimoto

doi:10.3934/dcds.2013.33.4095

Article Contents

2013, Volume 33, Issue 9: 4095-4122. Doi: 10.3934/dcds.2013.33.4095

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Resonant decomposition and the $I$-method for the two-dimensional Zakharov system

Nobu Kishimoto^1,

1.
Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502

Received: March 2012

Revised: December 2012

Published: September 2013

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Abstract

The initial value problem of the Zakharov system on a two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some `resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).

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Mathematics Subject Classification: Primary: 35Q55.

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References

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