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

建议DMS-04006627标题：非线性发展方程解的局部正则性和长时间行为PI：伊戈尔·罗德尼安斯基，普林斯顿大学摘要本建议的重点是研究爱因斯坦方程解的局部正则性和整体行为。局部行为被编码在基于L2的Sobolv空间的尺度上的局部正则性的传播中。我们将攻击L2曲率猜想，它断言解，度量g，在爱因斯坦真空方程中，Ric(G)=0可以局部扩张，只要它的曲率张量在L2中有界。这个问题超出了标准的基于傅立叶分析的方法的应用范围，因此需要开发新的分析工具。整体行为将在波坐标规范中Minkowski空间的稳定性问题的背景下被研究。这个特殊的规范被用来构造后Minkowski近似，我们希望提供一个额外的洞察这些展开式的有效性。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体演化的主要经典描述。产生于广义相对论的数学和物理现象的研究具有根本性的重要性。尽管对该学科的物理理解已经取得了迅速的进展，并产生了一些杰出的猜想，但严格的数学图景还没有出现。后者在很大程度上是由于爱因斯坦方程的高度非线性性质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，几何和偏微分方程式之间的接口。

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.