Global small amplitude solutions for two-dimensional nonlinear Klein–Gordon systems in the presence of mass resonance

Kawahara, Yuichiro; Sunagawa, Hideaki

doi:10.1016/j.jde.2011.04.001

Cited by 41 publications

(38 citation statements)

References 10 publications

(18 reference statements)

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“…This work simplified and generalized (by including resonant cases) the earlier works by Tsutsumi (1995, 1996), Tsutsumi (2003a,b) and Sunagawa (2003Sunagawa ( , 2004. See also Katayama, Ozawa, and Sunagawa (2012) for the algebraic characterization of the null condition and the asymptotic behavior of solutions, as well as Kawahara and Sunagawa (2011) for a condition weaker than the null condition.…”

Section: Further References On Earlier Workmentioning

confidence: 82%

The Hyperboloidal Foliation Method

LeFloch

2014

Series in Applied and Computational Mathematics

181

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ii iii iv PrefaceThe Hyperboloidal Foliation Method presented in this monograph is based on a p3`1q-foliation of Minkowski spacetime by hyperboloidal hypersurfaces. It allows us to establish global-in-time existence results for systems of nonlinear wave equations posed on a curved spacetime and to derive uniform energy bounds and optimal rates of decay in time. We are also able to encompass the wave equation and the Klein-Gordon equation in a unified framework and to establish a well-posedness theory for nonlinear wave-Klein-Gordon systems and a large class of nonlinear interactions.The hyperboidal foliation of Minkowski spactime we rely upon in this book has the advantage of being geometric in nature and, especially, invariant under Lorentz transformations. As stated, our theory applies to many systems arising in mathematical physics and involving a massive scalar field, such as the Dirac-Klein-Gordon system. As it provides uniform energy bounds and optimal rates of decay in time, our method appears to be very robust and should extend to even more general systems.We have built upon many earlier studies of nonlinear wave equations or Klein-Gordon equations, especially by Sergiu Klainerman, Demetri Christodoulou, Jalal Shatah, Alain Bachelot, and many others. The coupling of nonlinear wave-Klein-Gordon systems was first understood by Soichiro Katayama who succeeded to establish an existence theory of such systems.Importantly, in developing the Hyperboloidal Foliation Method, we were inspired by earlier work on the Einstein equations of general relativity by Helmut Friedrich, Vincent Moncrief, and Anil Zenginoglu.We are very grateful Soichiro Katayama for observations he made to the authors on a preliminary version of this monograph.Last but not least, the authors are very grateful to their respective families for their strong support.

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Section: Further References On Earlier Workmentioning

confidence: 82%

The Hyperboloidal Foliation Method

LeFloch

2014

Series in Applied and Computational Mathematics

181

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“…For Klein-Gordon equation, [18] combined the normal form transform developed in [12] and the vector field method from [11] and established the global existence for arbitrary quadratic nonlinearities in the case of single equation and "non-massresonance" system. Then [19] regarded the case with mass-resonance.…”

Section: Objectivementioning

confidence: 99%

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

2020

Nonlinear Analysis

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In this article we will develop some techniques aimed at the strong couplings in twodimensional wave-Klein-Gordon system. We distinguish the roles of different type of decay factors and develop a method which permits us to "exchange" one type of decay into the other. Then a global existence result of a model problem is established. We also give a sketch of the Klein-Gordon-Zakharov model system and establish the associate global existence result. * The present work belongs to a research project "Global stability of quasilinear wave-Klein-Gordon system in 2 + 1 space-time dimension" (11601414), supported by NSFC.

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“…In the case of one-dimensional cubic nonlinear Schrödinger equations, similar structural conditions have been considered in [37], [26], [19], [27], [36], [21], etc. Analogous consideration for quadratic nonlinear Klein-Gordon systems can be found in [10], [28], [25]. However, as far as the authors know, there are no previous papers which concern quadratic derivative nonlinear Schrödinger systems from this viewpoint.…”

Section: Resultsmentioning

confidence: 99%

Null Structure in a System of Quadratic Derivative Nonlinear Schrödinger Equations

Ikeda

Katayama

Sunagawa

2014

Ann. Henri Poincaré

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Abstract:We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.

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Global small amplitude solutions for two-dimensional nonlinear Klein–Gordon systems in the presence of mass resonance

Cited by 41 publications

References 10 publications

The Hyperboloidal Foliation Method

The Hyperboloidal Foliation Method

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

Null Structure in a System of Quadratic Derivative Nonlinear Schrödinger Equations

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