FRG: Collaborative Research: Long-Term Dynamics of Nonlinear Dispersive and Hyperbolic Equations: Deterministic and Probabilistic Methods

本课题主要研究与偏微分方程解的长期动力学有关的几个问题。这种类型的方程是建模和模拟一系列现象的中心，包括水波、光的传播、等离子体的行为和引力。研究人员是在偏微分方程研究的不同方面都有经验的资深研究人员；他们将在色散方程和双曲型方程的解的研究中的四个主要研究方向上发挥互补的专业知识。非线性偏微分方程解的四个方面都是理解重要的非线性、色散和双曲型方程解的长时间动力学这一总目标的表现。这个项目是由来自四个机构的五名高级研究人员共同努力的，他们在偏微分方程的各个领域都有研究经验。研究人员对色散和双曲型偏微分方程都很感兴趣，并将从几个不同的方向合作研究正在研究的问题。该项目主要集中在色散和双曲型方程解的研究中的四个主要研究方向：(1)临界半线性演化的分析，特别是扩展的II型解的“孤子解猜想”；(2)某些拟线性色散模型的整体解的构造，如水波模型、等离子体模型和晶体光学；(3)广义相对论中的稳定性问题，受突出的Kerr非线性稳定性猜想的启发；(4)与随机化数据相对应的解的长期动力学。非线性偏微分方程的这四个方面都是理解解的长期动力学的总体目标的表现。虽然确定了该项目的具体主要问题，但这是一个非常活跃的领域，预计该项目将在调查过程中确定和研究其他重要问题。该项目还将加强美国领先数学中心的研究人员之间的联系和合作，通过学生和博士后研究人员的积极参与来促进培训。该项目所产生的研究将在高级研究生课程、调查文章和专著中广泛传播。

This project focuses on several problems connected with the long-term dynamics of solutions of partial differential evolution equations. Equations of this type are central to the modeling and simulation of a wide range of phenomena, including water waves, propagation of light, behavior of plasmas, and gravitation. The investigators are senior researchers with experience in different aspects of the study of partial differential equations; their complementary expertise will be brought to bear on four main research directions in the study of solutions of dispersive and hyperbolic equations. The four aspects of nonlinear partial differential equations are all manifestations of the general goal of understanding the long time dynamics of solutions to important nonlinear dispersive and hyperbolic equations. This project is a collective effort of a group of five senior researchers from four institutions with research experience in various areas of partial differential equations. The investigators have been interested in both dispersive and hyperbolic partial differential equations and will collaborate to approach the problems under study from several different directions. The project centers on four main research directions in the study of solutions of dispersive and hyperbolic equations: (1) the analysis of critical semilinear evolutions, with special emphasis on the "soliton resolution conjecture" for extended type II solutions, (2) construction of global solutions of certain quasilinear dispersive models, such as water wave models, plasma models, and crystal optics, (3) stability problems in General Relativity, motivated by the outstanding Kerr nonlinear stability conjecture, and (4) long-term dynamics of solutions corresponding to randomized data. These four aspects of nonlinear partial differential equations are all manifestations of the general goal of understanding the long time dynamics of solutions. While specific major problems are identified for the project, this is a very active field, and it is anticipated that this project will identify and study other important problems during the course of the investigation. The project will also enhance links and collaborations among researchers at leading mathematical centers in the U.S., promoting training through active research involvement of students and postdoctoral researchers. Research resulting from this project will be disseminated widely in advanced graduate courses, survey articles, and monographs.