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

这个为期三年的奖项支持美国和法国在应用数学领域的合作研究。该提案涉及美国和法国的两个研究小组，分别由乔治亚理工学院的杰克·黑尔和南巴黎大学的吉纳维夫·劳格尔领导。研究者建议研究无限维动力物理系统的定性性质。这样的系统通常用偏微分方程（PDE）或无穷常微分方程（ODE）或无穷维集的映射来描述。利用这些方法和其他与数学物理和量子系统相关的方法，研究人员将研究一些具体问题：(1)耗散偏微分方程定义的流动性质；(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)动力学系统的相空间扩散特性和薛定谔算子谱与驱避器谱之间的关系。佐治亚理工学院的小组通过研究常微分方程和图来解决动力系统的问题。他们为这次合作带来了功能分析和数学物理方面的专业知识。Orsay小组由应用于动力系统问题的偏微分方程专家组成。他们精通算子理论和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。

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.