U.S.-France Cooperative Research: Qualitative Properties inDynamics and Mathematical Physics

这项为期三年的奖项支持美法在应用数学方面的合作研究。该提案涉及美国和法国的两个研究小组，由佐治亚理工学院的杰克·黑尔和巴黎大学南方分校(Orsay)的吉纳维芙·劳格尔领导。研究人员建议研究无限维动力、物理系统的定性性质。这类系统通常用偏微分方程组(PDE)或无穷常微分方程组(ODE)或某些无穷维集的映射来描述。利用这些方法以及与数学物理和量子系统有关的其他方法，研究人员将研究一些具体的问题：(1)耗散偏微分方程定义的流动的性质；(2)扩展动力系统中的湍流(时空混沌)、相干结构和间歇性的概念；(3)相空间扩散性质之间的关系，以及薛定谔算符的谱与动力系统的排斥子之间的关系。佐治亚理工学院的研究小组通过研究常微分方程式和地图来解决动力系统问题。他们为这次合作带来了泛函分析和数学物理方面的专业知识。奥赛小组由应用于动力系统问题的偏微分方程式方面的专家组成。他们精通算子论和C*-代数。他们的共同努力将把动力系统理论扩展到偏微分方程式。

This three-year award supports U.S.-France cooperative research in applied mathematics. The proposal involves two research groups in the U.S. and France led by Jack Hale of the Georgia Institute of Technology and Genevieve Raugel of the University of Paris, South (Orsay). The investigators propose to study the qualitative properties of infinite dimensional dynamical, physical systems. Such systems are usually described by partial differential equations (PDE) or by infinite systems of ordinary differential equations (ODE) or by maps of some infinite dimensional sets. Utilizing these methods and others related to mathematical physics and quantum systems, the investigators will investigate some concrete problems: (1) the properties of flows as defined by dissipative PDE; (2) notions of turbulence (space- time chaos), coherent structures, and intermittency in extended dynamical systems; and (3) relationships between the properties of diffusion in phase space and between the spectrum of the Schrodinger operator and repellers of dynamical systems. The Georgia Tech group approaches problems of dynamical systems through the study of ordinary differential equations and maps. They bring to this collaboration expertise in functional analysis and mathematical physics. The Orsay group consists of specialists in partial differential equations as applied to problems in dynamical systems. They are well versed in operator theory and C*-algebras. Their combined efforts will extend the theory of dynamical systems to partial differential equations.