Mathematical Sciences: On Some Problems Which Arise in the Study of Invariant Measures of Nonlinear Evolutionary Equations.

数学科学：非线性演化方程不变测度研究中出现的一些问题。

• 批准号: 9501002
• 负责人: Kirill Vaninsky
• 金额: $ 4.5万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501002&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Arise

PI: Vaninsky DMS-9501002 Three problems of statistical mechanics of one dimensional systems will be investigated. The first is a construction of dynamics for the nonlinear Schrodinger and modified Korteveg-de Vries type equations on the entire line for rough initial data from the support of translation-invariant Gibbsian distribution. The second is construction of the action-angle variables for integrable equations on the entire line for such initial data. The third is the soliton gas conjecture. Nonequilibrium statistical mechanics studies properties of dynamical systems describing big or infinite systems of interacting particles. The Hamiltonian structure of the equations of motion allows one to define in the phase space the so-called Gibbsian measures, invariant under the dynamics. These measures are usually constructed form the basic Hamiltonian and other classical conserved quantities such as the number of particles, integrals of momentum and angular momentum. Statistical mechanics is based on Gibb's postulate which states that this finite parameter family of measures exhausts the class of all translation invariant measures with weak long-range dependence.

PI：Vaninsky DMS-9501002 将研究一维系统统计力学的三个问题。 第一个是在平移不变吉布斯分布的支持下为粗略初始数据构建非线性薛定谔动力学和整条线上修正的 Korteveg-de Vries 型方程。 第二个是在此类初始数据的整条线上构造可积方程的作用角变量。 第三个是孤子气体猜想。 非平衡统计力学研究描述相互作用粒子的大或无限系统的动力系统的性质。 运动方程的哈密顿结构允许人们在相空间中定义所谓的吉布斯测度，在动力学下保持不变。 这些度量通常是根据基本哈密顿量和其他经典守恒量（例如粒子数、动量积分和角动量）构建的。 统计力学基于吉布假设，该假设指出，这种有限参数测度族穷尽了所有具有弱远程依赖性的平移不变测度的类别。