On the Cauchy Problem for Energy Critical Self-Gravitating Wave Maps
On the Cauchy Problem for Energy Critical Self-Gravitating Wave Maps
复制标题
关于能量临界自引力波图的柯西问题
DOI：
10.17169/refubium-6002
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
N. Gudapati
中科院分区：
文献类型：
--
作者：
N. Gudapati
这项工作是柯西问题的临界波映射耦合到爱因斯坦的广义相对论方程。本文的主要结果是证明了爱因斯坦等变波映象系统在柯西演化过程中能量不集中。证明中的一个关键因素是使用这样一个事实，即爱因斯坦等变波映射系统在无穷远处的几何质量在演化过程中是守恒的。然而，这一观测结果有一些微妙的局部含义，已被用来估计当地的能量。例如，我们构造了一个无发散向量场，它给出了能量在任意点的过去零锥中的单调性。此外，这个向量也被用来证明能量不会集中在远离轴的区域流形。随后，我们估计了Morawetz矢量在截短的过去零锥上的发散，证明了动能不集中。最后，假设目标流形满足Grillakis条件，我们继续证明临界爱因斯坦等变波映射系统的能量不集中。保持跟踪的各种数量的波映射相对于不断发展的零几何的背景流形是一个反复出现的主题，在整个过程中的这项工作。
除了纯粹的数学兴趣之外，研究临界自引力波图的动机是它们自然地出现在爱因斯坦的3+1广义相对论方程中。因此，研究临界自引力波图可能是理解爱因斯坦方程难以捉摸的全局行为的一种富有成效的方式。这项工作是这一努力的一个步骤。
This work is on the Cauchy problem for critical wave maps coupled to Einstein's equations of general relativity. The main result of this work is the proof that the energy of the Einstein-equivariant wave map system does not concentrate during the Cauchy evolution. A key ingredient in the proof is the use of the fact that geometric mass at infinity of the Einstein-equivariant wave map system is conserved during the evolution. However, this observation has some subtle local implications which have been used to estimate the energy locally. For instance, we construct a divergence-free vector field which gives monotonicity of energy in the past null cone of any point. In addition, this vector has also been used to prove that the energy does not concentrate away from the axis of the domain manifold. Later, estimating the divergence of a Morawetz vector on a truncated past null cone, we prove that the kinetic energy does not concentrate. Finally, assuming that the target manifold satisfies the Grillakis condition, we proceed to prove the non-concentration of energy for the critical Einstein-equivariant wave map system. Keeping track of various quantities of wave map relative to the evolving null geometry of the background manifold is a recurring theme throughout the course of this work.
Apart from a purely mathematical interest, the motivation to study critical self-gravitating wave maps is that they occur naturally in 3+1 Einstein's equations of general relativity. Therefore, studying critical self-gravitating wave maps could be a fruitful way of understanding the ever elusive global behaviour of Einstein's equations. This work is a step in this endeavour.