Nonlinear partial differential equations and applications

非线性偏微分方程及其应用

• 批准号: 0244787
• 负责人: Panagiotis Souganidis
• 金额: --
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2007-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244787&HistoricalAwards=false
• 关键词: Nonlinear; partial; differential; equations; applications

PI: Takis Souganidis, University of Texas, AustinDMS-0244787ABSTRACTThe PI proposes to continue his program to develop methods to study nonlinear, parabolic/elliptic and hyperbolic, deterministic and stochastic partial differential equations (pde) arising as models in areas like continuum and statistical physics, biology, ecology, engineering, etc. The emphasis of this proposal is on the development of (rigorous) theories for (i) weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolic stochastic partial differential equations, and(ii) homogenization (averaging) of nonlinear, parabolic/elliptic and hyperbolic partial differential equations in spatio-temporal randomly heterogeneous media. The randomness is typically associated with singular dependence on the state variables and lack of compactness. Overcoming both issues requires the development of new methods and tools.Nonlinear stochastic pde are very often used to model complex phenomena, like turbulent combustion, nucleations, phase transitions at low temperature, etc., It is therefore self-evident that there is a pressing need for the development of a mathematical theory for such equations. The theory of stochastic viscosity solutions, which is been developed by the PI and Lions, is of fundamental importance, since it allows for the study of a completely new and, at the same time, very rich class of nonlinear stochastic pde arising in many of the aforementioned contexts. As this theory becomes more available, known and developed, it is expected that it will play an important role also in applications, since it will provide the necessary tools to analyze previously intractable models.The mathematical modeling of phenomena with many scales at the microscopic level, like turbulence, front propagation in phase transitions, spatial ecology, polymer growth, composite and perforated materials in mechanics, often necessitates the use of random pde with oscillating random parameters. At the macroscopic level these parameters "average out" and the phenomena are then described by an averaged (homogenized) equation. This leads naturally to questions of homogenization of nonlinear pde in spatio-temporal stationary ergodic settings, which, roughly speaking, are the most general settings where "self-averaging" is to be expected.

主要研究者：Takis Souganwei，德克萨斯大学，AustinDMS-0244787摘要PI建议继续他的计划，以发展研究非线性，抛物/椭圆和双曲，确定性和随机偏微分方程（PDE）的方法，这些方程作为连续统和统计物理，生物学，生态学，工程学等领域的模型出现。本建议的重点是发展（一）弱（弱）的（严格）理论（随机粘性）完全非线性的解决方案，（退化）抛物随机偏微分方程，（ii）均匀化时空随机非均匀介质中的非线性、抛物/椭圆和双曲型偏微分方程。随机性通常与对状态变量的奇异依赖和缺乏紧凑性相关联。克服这两个问题需要开发新的方法和工具。非线性随机偏微分方程经常被用来模拟复杂的现象，如湍流燃烧，成核，低温相变等，因此，不言而喻，迫切需要为这种方程发展一种数学理论。理论的随机粘性的解决方案，这是由PI和狮子，是具有根本的重要性，因为它允许研究一个全新的，同时，非常丰富的一类非线性随机偏微分方程中产生的许多上述情况。随着这一理论的日益普及、被人们所熟知和不断发展，预计它将在应用中发挥重要作用，因为它将为分析以前难以处理的模型提供必要的工具。微观层次上具有许多尺度的现象的数学建模，如湍流、相变中的前沿传播、空间生态学、聚合物生长、力学中的复合材料和穿孔材料，通常需要使用具有振荡随机参数的随机PDE。在宏观层面上，这些参数“平均”，然后通过平均（均匀化）方程描述这些现象。这自然会导致时空平稳遍历环境中非线性偏微分方程的均匀化问题，粗略地说，这是最普遍的预期"自平均"的环境。