Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems
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- by James Colliander, Justin Holmer and Nikolaos Tzirakis PDF
- Trans. Amer. Math. Soc. 360 (2008), 4619-4638 Request permission
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.References
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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
- MR Author ID: 759238
- 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
- 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. - © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 4619-4638
- MSC (2000): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9947-08-04295-5
- MathSciNet review: 2403699