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.

拟议的研究策略术的非线性部分微分方程和应用摘要是应用科学中最具挑战性的问题之一是对具有许多自由度（量表）的现象的建模，这些现象有望具有一些平均（巨镜），也许是随机的，是随机的。自然界中问题的多尺度和复杂性通常需要使用随机培养基。在许多应用程序中，也只有``统计''（随机）而不是``eactry''（确定性）信息也很常见。此外，物理相对数量的波动的建模导致方程``奇异''（白噪声类型）依赖于某些变量。在这种情况下，随机均质化和随机部分微分方程成为自然数学对象。随机性与对状态变量的奇异依赖性和缺乏紧凑性有关，这两个事实引起了具有挑战性的数学问题。克服它们需要开发新方法和技术。在生物学中，分子量表的最新实验导致了新的复杂数学模型。需要新的工具和思想来进一步研究这些问题，并确定参数的所有相关制度/量表，这会影响实验观察到的行为。基于（i）完全非线性，（退化）抛物性抛物面PDE的弱（随机粘度）溶液的理论发展，（ii）非线性，抛物线/椭圆形和双曲线PDE的均质化在时空 - 周期性媒体中的抛物性/椭圆形和双曲线PDE的均质（III）（iii）的属性（常规性估计），以及VISC估计的数字估计，VISCITION的差异（差异）。生物学。