Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用

• 批准号: 0555826
• 负责人: Panagiotis Souganidis
• 金额: $ 20.95万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555826&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Applications

Nonlinear Partial Differential Equations and Applications Abstract of Proposed ResearchPanagiotis E Souganidis One of the most challenging problems in applied sciences is the modeling of phenomena with many degrees of freedom (scales), which are expected to have some averaged (macroscopic), perhaps random, behavior. The multi-scales and complexity of the problems in nature often necessitate the use of random media. In many applications, it is also common to have only ``statistical'' (random) and not ``exact'' (deterministic) information. In addition, the modeling of the fluctuations of the physically relative quantities leads to equations with ``singular'' (white noise type) dependence on some of the variables. In this context, random homogenization and stochastic partial differential equations become the natural mathematical objects. The randomness is associated with singular dependence on the state variables and lack of compactness, two facts that give rise to challenging mathematical problems. Overcoming them requires the development of new methods and techniques. In biology, recent experiments at the molecular scale have led to new sophisticated mathematical models. Novel tools and ideas are needed to study these problems further and to identify all the relevant regimes/scales of the parameters, which affect the experimentally observed behavior.The PI proposes to continue his program to develop methods to study nonlinear deterministic (parabolic/elliptic and hyperbolic) deterministic and stochastic partial differential equations arising in models in areas such as continuum and statistical physics, biology, engineering, etc. The emphasis of the proposal is on the development of theories for (i) weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolicstochastic pde, (ii) the homogenization of nonlinear, parabolic/elliptic and hyperbolic pde in spatio-temporal random media, (iii) the study of properties (regularity, error estimates) of viscosity solutions, and (iv) the analysis of some models in mathematical biology.

非线性偏微分方程和应用摘要建议researchPanagiotis E Souganeville应用科学中最具挑战性的问题之一是建模的现象与许多自由度（尺度），这是预期有一些平均（宏观），也许随机，行为。自然界中问题的多尺度性和复杂性使得随机介质的使用成为必然。在许多应用中，通常只有“精确”（随机）信息而没有“精确”（确定）信息。此外，对物理上相对量的波动进行建模，导致方程对某些变量具有“奇异”（白色噪声类型）依赖性。在这种背景下，随机均匀化和随机偏微分方程成为自然的数学对象。随机性与对状态变量的奇异依赖和缺乏紧凑性有关，这两个事实引起了具有挑战性的数学问题。克服这些问题需要发展新的方法和技术。在生物学中，最近在分子尺度上的实验已经导致了新的复杂的数学模型。需要新的工具和想法来进一步研究这些问题，并确定影响实验观察行为的所有相关参数/范围。PI建议继续他的计划，以开发研究非线性确定性的方法（抛物线/椭圆和双曲线）确定性和随机偏微分方程产生的模型，如连续和统计物理，生物学，工程，建议的重点是发展理论，以（i）弱（随机粘性）完全非线性的解决方案，（ii）时空随机介质中非线性、抛物/椭圆和双曲型偏微分方程的均匀化，（iii）性质的研究（正则性，误差估计）的粘度解决方案，和（iv）分析的一些数学生物模型。