Collisionless matter in general relativity

广义相对论是爱因斯坦于1915年首次提出的一种引力理论。尽管已经有100多年的历史了，但在理解这个理论及其控制方程——爱因斯坦方程——方面仍然存在许多巨大的数学挑战，恰当地看，爱因斯坦方程是一个非线性的双曲偏微分方程系统。这个理论的一个特别引人注目的方面，与之前的牛顿引力理论形成鲜明对比，是它在真空中是非平凡的，即在没有任何物质的情况下。近年来，真空条件下的爱因斯坦方程研究取得了巨大进展。然而，在有物质存在的情况下，进展要慢得多。拟议的研究试图弥合这一差距。广义相对论的许多数学研究都是由所谓的宇宙审查猜想推动的，该猜想首先由彭罗斯提出，它假定理论中出现的任何奇点都必须具有某些理想的性质。在偏微分方程的语言中，这些猜想可以看作是全局存在性和全局唯一性的陈述。一个卓有成效的研究方向，确实构成了迄今为止关于猜想的大部分进展，包括展示接近某些明确的固定解的制度，其中猜想成立。在物质存在的情况下，进展缓慢是因为人们不仅要理解真空中产生的困难和物质方程单独产生的困难，还要理解由于这些困难的耦合而产生的新困难。这一建议涉及一种特别简单的物质，尽管如此，它仍然被认为是物理的，并在广义相对论中表现出广泛的新现象，由弗拉索夫方程描述。

General relativity is a theory of gravity first proposed by Einstein in 1915. Despite being over 100 years old, many great mathematical challenges remain in understanding the theory and its governing equations -- the Einstein equations -- which, appropriately viewed, are a nonlinear system of hyperbolic partial differential equations. One particularly striking aspect of the theory, which is in sharp contrast to the preceding Newtonian theory of gravity, is that it is nontrivial in the vacuum, i.e. in the absence of any matter. Tremendous progress has been made on the Einstein equations in vacuum recent years. Progress in the presence of matter, however, has been considerably slower. The proposed research attempts to bridge this gap.Much of the mathematical study of general relativity is driven by the so called cosmic censorship conjectures, first formulated by Penrose, which posit that any singularities arising in the theory must have certain desirable properties. In the language of partial differential equations the conjectures can be viewed as statements of global existence and global uniqueness. A fruitful direction of study, which indeed constitutes much of the progress to date on the conjectures, involves exhibiting regimes, close to certain explicit stationary solutions, in which the conjectures hold.The slow progress in the presence of matter is due to the fact that one has to not only understand difficulties arising in the vacuum and difficulties arising from the matter equations alone, but also the new difficulties which arise due to the coupling of these difficulties. This proposal concerns a particularly simple type of matter, which nonetheless is regarded as physical and exhibits a wide range of new phenomena in general relativity, described by the Vlasov equation.