Local Regularity and Long Time Behavior of Solutions on Non-Linear Evolution Equations

提案DMS-04006627标题：非线性演化方程解的局部正则性和长时间行为PI：Igor Rodnianski，Princeton大学摘要本文的重点是研究Einstein方程解的局部正则性和整体行为，局部行为被编码为局部正则性在基于L2的Sobolev空间中的传播，我们将攻击L2曲率猜想，它断言一个解，度量g，爱因斯坦真空方程Ric（g）= 0的曲率张量只要在L2中有界，就可以局部延拓.这个问题超出了标准的Fourier分析方法的应用范围，因此需要发展新的分析工具.本文将结合波坐标规范下Minkowski空间的稳定性问题来研究它的整体行为.这一结果表明，在波坐标规范下，Minkowski空间的稳定性是一个很好的研究对象.一个特殊的规范在物理学中被用来构造后闵可夫斯基近似，我们希望提供一个额外的洞察力，这些扩展的有效性。对于不满足标准零条件的方程，稳定性问题属于小数据问题解的整体存在性范畴，广义相对论的爱因斯坦方程提供了物理时空连续统演化的主要经典解释。 对广义相对论中数学和物理现象的研究具有重要的基础性，虽然对这一问题的物理认识已经取得了迅速的进展，并产生了许多杰出的成果，但严格的数学图象尚未出现.后者在很大程度上是由于爱因斯坦方程的高度非线性和缺乏处理它的数学工具。一个令人满意的数学方法的发展取决于分析，几何和偏微分方程之间的接口。

Proposal DMS-04006627Title: Local regularity and long time behavior of solutions of nonlinear evolution equationsPI: Igor Rodnianski, Princeton UniversityABSTRACTThe focus of this proposal is the study of the local regularity andglobal behavior of solutions of the Einstein equations.The local behavior is encoded in the propagation of local regularityin the scale of the L2 based Sobolev spaces.We will attack the L2 curvature conjecture, which assertsthat a solution, metric g, of the Einstein vacuum equations Ric(g)=0can be locally extended as long as its curvature tensor is bounded in L2.The problem is beyond the range of application of the standard Fourier analysisbased methods and thus will require development of new analytic tools.The global behavior will be studied in the context of the problemof stability of Minkowski space in the wave coordinate gauge.This particular gauge is used in physics to construct the post-Minkowskianapproximations and we hope to provide an additional insight into validityof these expansions. The stability problem belongs to the category of globalexistence for solutions with small data problems for the equations not satisfyingthe standard null condition.The Einstein equations of General Relativity provide the main classical descriptionof evolution of the physical space-time continuum. The study of mathematical andphysical phenomena arising in General Relativity is of the fundamental importance.While the physical understanding of the subject has made rapid advancement and generated a number of outstanding conjectures the rigorous mathematical picture isyet to emerge. The latter is to a large extent due to a highly nonlinearnature of the Einstein equations and a lack of the mathematical tools to deal with it.The development of a satisfactory mathematical approach lies on the interface betweenAnalysis, Geometry and Partial Differential Equations.