Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用研究摘要Panagiotis E Souganidis是应用科学中最具挑战性的问题之一，它是对具有多个自由度(尺度)的现象的建模，这些现象被期望具有某种平均的(宏观的)，可能是随机的行为。自然界中问题的多尺度和复杂性往往需要使用随机媒体。在许多应用中，只有“统计”(随机)而没有“精确”(确定性)信息也是很常见的。此外，对物理相对量的涨落进行建模会导致方程对某些变量具有“奇异”(白噪声类型)依赖关系。在此背景下，随机齐次化和随机偏微分方程组成为自然而然的数学对象。随机性与对状态变量的单一依赖和紧凑性的缺乏有关，这两个事实导致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。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。需要新的工具和思想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的状态/标度。Pi建议继续他的计划，以发展方法来研究在连续介质和统计物理、生物、工程等领域的模型中出现的非线性确定性(抛物线/椭圆型和双曲型)确定性和随机偏微分方程组。建议的重点是发展(I)完全非线性、(退化)抛物型随机偏微分方程组的弱(随机粘性)解，(Ii)时空随机介质中非线性、抛物型/椭圆型和双曲型偏微分方程组的齐次化，(3)粘性解的性质(正则性、误差估计)的研究；(4)数学生物学中一些模型的分析。

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.