Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用

• 批准号: 0901802
• 负责人: Panagiotis Souganidis
• 金额: $ 50.8万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901802&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Applications

SouganidisThe modeling of multi-scale phenomena necessitates the use of random media (periodicity is a rather restrictive structure for many applications) and requires the study of averaged (mesoscopic and macroscopic) behaviors. For complex phenomena, it is also very common to have only \statistical? (random) and not \exact? (deterministic) information. In addition, incorporating the fluctuations of several physical quantities leads to equations with \singular? (white noise type) dependence on some of the variables. In this context, random homogenization and stochastic pde become the natural mathematical objects. The randomness is associated with singular dependence on the state variables and lack of compactness both giving rise to challenging mathematical problems. Overcoming them requires the development of new methods and techniques. Viscosity solutions of nonlinear ¯rst- and second-order pde have become a classical tool for the study of many applications. It is therefore important to improve the understanding of their qualitative properties. 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 affecting the experimentally observed behavior. The PI proposes to continue his program to develop methods to study nonlinear deterministic and stochastic pde arising in continuum and statistical physics, biology, engineering, etc.. The emphasis of the proposal is on (i) the development of theories for weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolic stochastic pde, and the homogenization of nonlinear, parabolic/elliptic and hyperbolic pde in spatiotemporal random media, (ii) the study of problems related to viscosity solutions (rates of convergence, averaging, large deviations in infinite dimensions, etc.), and (iii) the analysis of models for motor and concentration effects in mathematical biology. First- and second-order, stochastic pde and stochastic homogenization arise in models for a wide variety of phenomena and applications including turbulence, phase transitions and front propagation in random media with/out random velocities, nucleations in physics, macroscopic limits of particle systems, stochastic control theory, stochastic control with partial observations, financial mathematics, etc.. The theory of stochastic viscosity solutions is important. It allows for the study of a completely new class of fully nonlinear stochastic pde. As the theory develops further, it is expected that it will play a crucial role in applied areas by providing the necessary tools to analyze previously intractable models. There has been resurgence of interest in homogenization in random media. The novel tools proposed to be developed are expected to become the standard methodology in the field. Providing a unified (analytic) treatment, based on viscosity solutions, to a class of large deviation problems in infinite dimensions will lead to a better understanding of an important area with applications to particle systems, random matrices, phase transitions, etc.. In mathematical biology, the proposed work is expected to enhance the understanding of concrete phenomena like bio-motor behavior and concentration effects.

索加尼斯主义多尺度现象的建模需要使用随机介质(对于许多应用来说，周期性是一个相当有限制性的结构)，并需要研究平均(中观和宏观)行为。对于复杂的现象，只有统计上的？(随机的)和不精确的？(确定性)信息。此外，将几个物理量的涨落结合在一起，得到了具有奇异的？(白噪声类型)依赖于某些变量。在此背景下，随机齐次化和随机偏微分方程成为自然而然的数学对象。随机性与对状态变量的奇异依赖和紧凑性的缺乏有关，这两者都带来了具有挑战性的数学问题。克服这些问题需要开发新的方法和技术。一阶和二阶非线性偏微分方程组的粘性解已成为研究许多应用的经典工具。因此，重要的是要提高对其定性性质的理解。在生物学方面，最近在分子尺度上的实验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到的行为的参数的所有相关制度/尺度。PI建议继续他的计划，开发研究连续介质和统计物理、生物、工程等领域中出现的非线性确定性和随机PDE的方法。该建议的重点是(I)发展完全非线性、(退化)抛物型随机PDE的弱(随机粘性)解的理论，以及时空随机介质中非线性、抛物型/椭圆型和双曲型PDE的齐次化，(Ii)与粘性解有关的问题的研究(收敛速度、平均速度、无限维大偏差等)，以及(Iii)数学生物学中运动和浓度效应的模型分析。一阶和二阶随机偏微分方程组和随机齐次化出现在各种现象和应用中，包括湍流、相变和随机介质中的前沿传播、物理成核、粒子系统的宏观极限、随机控制理论、带部分观测的随机控制、金融数学等。随机粘性解的理论是重要的。它允许研究一类全新的完全非线性随机偏微分方程组。随着该理论的进一步发展，预计它将通过提供必要的工具来分析以前难以处理的模型，从而在应用领域发挥关键作用。人们对随机介质中的同质化重新产生了兴趣。拟议开发的新工具有望成为该领域的标准方法。基于粘性解对一类无限维大偏差问题进行统一的(解析)处理，将有助于更好地理解粒子系统、随机矩阵、相变等领域的重要应用。在数学生物学方面，这项拟议的工作有望加强对生物运动行为和注意力效应等具体现象的理解。