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

题目：非线性演化方程解的局部正则性和长时间行为[j]: Igor Rodnianski，普林斯顿大学[j] .摘要本文主要研究爱因斯坦方程解的局部正则性和全局行为。局部行为在基于L2 Sobolev空间尺度的局部正则性传播中进行编码。我们将攻击L2曲率猜想，它断言爱因斯坦真空方程Ric(g)=0的解，度规g可以局部扩展，只要它的曲率张量在L2中有界。这个问题超出了基于标准傅立叶分析方法的应用范围，因此需要开发新的分析工具。本文将在波浪坐标规范中闵可夫斯基空间稳定性问题的背景下研究其全局行为。这种特殊的标准在物理学中用于构建后闵可夫斯基近似，我们希望为这些扩展的有效性提供额外的见解。该稳定性问题属于不满足标准零条件的小数据问题解的整体存在性问题。爱因斯坦广义相对论方程提供了物理时空连续体演化的主要经典描述。对广义相对论中出现的数学和物理现象的研究具有根本性的重要性。虽然对这一主题的物理理解已经取得了迅速的进展，并产生了许多杰出的猜想，但严格的数学图景尚未出现。后者在很大程度上是由于爱因斯坦方程的高度非线性性质和缺乏数学工具来处理它。一种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界面。

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.