The Hyperboloidal Foliation Method
The Hyperboloidal Foliation Method
复制标题
双曲面叶化法
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Yue Ma
中科院分区：
文献类型：
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作者：
P. LeFloch;Yue Ma
在这本专著中提出的双曲叶方法是基于闵可夫斯基时空的双曲超曲面的（3+1）叶化。它使我们能够建立一个弯曲的时空上的非线性波动方程系统的整体存在性结果，并推导出统一的能量边界和最佳衰减率的时间。我们也能够涵盖波动方程和Klein-Gordon方程在一个统一的框架，并建立一个适定性理论的非线性波动-Klein-Gordon系统和一个大类的非线性相互作用。我们在本书中所依赖的闵可夫斯基时空的双曲叶理具有几何性质的优点，特别是在洛伦兹变换下是不变的。如前所述，我们的理论适用于数学物理中出现的许多系统，并涉及大量的标量场，如狄拉克-克莱因-戈登系统。由于它提供了统一的能量界限和最佳的衰减率，我们的方法似乎是非常强大的，应该扩展到更一般的系统。
The Hyperboloidal Foliation Method presented in this monograph is based on a (3+1)-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 hyperboloidal foliation of Minkowski spacetime 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.