| 简介 |
The 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.
Philippe G. LeFloch and Yue Ma
Paris, September 2014 |
(创新港)
