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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems


Authors: James Colliander, Justin Holmer and Nikolaos Tzirakis
Journal: Trans. Amer. Math. Soc. 360 (2008), 4619-4638
MSC (2000): Primary 35Q55
Published electronically: April 11, 2008
MathSciNet review: 2403699
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Abstract: We prove low regularity global well-posedness for the 1d Zakharov system and the 3d Klein-Gordon-Schrödinger system, which are systems in two variables $ u:\mathbb{R}_x^d\times \mathbb{R}_t \to \mathbb{C}$ and $ n:\mathbb{R}^d_x\times \mathbb{R}_t\to \mathbb{R}$. The Zakharov system is known to be locally well-posed in $ (u,n)\in L^2\times H^{-1/2}$ and the Klein-Gordon-Schrödinger system is known to be locally well-posed in $ (u,n)\in L^2\times L^2$. Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the $ L^2$ norm of $ u$ and controlling the growth of $ n$ via the estimates in the local theory.


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Additional Information

James Colliander
Affiliation: Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4

Justin Holmer
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Nikolaos Tzirakis
Affiliation: Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4
Address at time of publication: Department of Mathematics, University of Illinois, 1409 Green St., Urbana, Illinois 61801

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04295-5
PII: S 0002-9947(08)04295-5
Keywords: Zakharov system, Klein-Gordon-Schr\"odinger system, global well-posedness
Received by editor(s): March 27, 2006
Received by editor(s) in revised form: April 17, 2006
Published electronically: April 11, 2008
Additional Notes: The first author was partially supported by N.S.E.R.C. Grant RGPIN 250233-03 and the Sloan Foundation.
The second author was supported by an NSF postdoctoral fellowship.
Article copyright: © Copyright 2008 American Mathematical Society