Nonlinear Partial Differential Equations and Applications

物理、社会科学和工程中许多现象的建模，如多孔介质、复合材料、湍流和燃烧、交通模型、犯罪传播、代理人模型等，都涉及由偏微分方程描述的异质介质。这些通常取决于许多参数，并且在小范围内随机变化。此外，可获得的资料（例如，用于天气预报的数据）往往不是精确的（确定性的），而是统计的（随机的），波动很大。在比异质性大得多的宏观尺度上，模型往往表现出比原模型简单得多的有效确定性行为。对这些数据取平均的过程称为均质化。从数学上讲，这意味着原来的随机问题被确定性问题所取代。当这种平均是不可能的，这是典型的情况下，当波动太强（野），有必要处理所谓的随机介质（随机偏微分方程），在空间和时间具有相当奇异的行为。随机平均和随机偏微分方程的数学研究都需要新颖的思想和新方法的发展，因为这两个主题都属于传统的平均和偏微分方程理论之外。另一个出现类似问题的新兴研究领域是数学生物学，在这个领域，分子尺度上的实验以及理论的进步，已经产生了新的、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，并验证影响实验观察和理论推测行为的参数的所有相关制度/尺度。该项目旨在发展研究随机均匀化、非线性随机偏微分方程的一般方法，并将其应用于前传播、相变和数学生物学。随机环境比周期性环境更普遍。后者基本上涉及某个方程的固定平移，而前者可以被认为涉及所有可能的（相关）方程。这导致了关于缺乏紧凑性的相当大的问题。因此，有必要提出新颖的论点，将受到审查的媒体的差异和随机结构结合起来。在这种情况下，方程是随机变量，特殊的依赖性表示方程在空间中被观察到的位置。首席研究员和他的合作者是第一个考虑在平稳遍历环境中的随机均匀化。这个项目的很大一部分致力于进一步发展这一理论。随机偏微分方程的系数具有非常奇异（布朗）的行为。在线性上下文中，这通常可以通过已知的方法来处理，例如经典的鞅方法。这种方法基于方程高阶部分的线性特征，因此不能用于非线性问题，因为非线性问题需要找到合适的替代解概念。这些，在一阶和二阶非线性方程的背景下，是由首席研究员和他的合作者引入的随机粘度和路径熵解。该项目的一部分是研究这些溶液的定性行为/性质。在数学生物学的背景下，首席研究员计划研究适应/选择模型以及发育生物学模型。前者涉及物种对全球变化的适应、昆虫对杀虫剂的抗性等问题。后者旨在开发模型来研究如何向增殖细胞提供位置信息，主要问题是尖锐和精确边界的形成和位置。

The modeling of many phenomena in the physical and social sciences and engineering, such as porous media, composite materials, turbulence and combustion, traffic models, spread of crime, agent models and others, involve heterogeneous media described by partial differential equations. These typically depend upon many parameters and vary randomly on a small scale. In addition, often the available information (e.g., data used in weather prediction) is not exact (deterministic) but statistical (random), with large fluctuations. On macroscopic scales that are much larger than the ones of the heterogeneities, the models often exhibit an effective deterministic behavior, which is much simpler than the original one. The process of averaging such data is known as homogenization. Mathematically, this means that the original random problem is replaced by a deterministic one. When this averaging is not possible, which is typically the case when the fluctuations are too strong (wild), it is necessary to deal with so-called stochastic media (stochastic partial differential equations), which have rather singular behavior in space and time. The mathematical study of both the stochastic averaging and the stochastic partial differential equations requires original ideas and the development of new methodologies, since both topics fall outside the traditional theories of averaging and partial differential equations. Another burgeoning area of research in which similar issues surface is mathematical biology, where experiments at the molecular scale, as well as theoretical advances, have led to new, sophisticated mathematical models. Novel tools and ideas are needed to study these problems further and to validate all the relevant regimes/scales of the parameters affecting the experimentally observed and theoretically conjectured behaviors. This project is directed at the development of general methodologies to study random homogenization, nonlinear stochastic partial differential equations, and applications to front propagation, phase transitions, and mathematical biology. Random environments are much more general than periodic ones. The latter basically involve fixed translations of a certain equation, whereas the former can be thought of as involving all possible (relevant) equations. This leads to considerable issues concerning the lack of compactness. It is therefore necessary to develop novel arguments that combine both the differential and random structures of the media under scrutiny. In this setting, the equation is the random variable and the special dependence signifies the location in space where the equation is observed. The principal investigator and his collaborators were the first to consider stochastic homogenization in stationary ergodic environments. A large part of the project is dedicated to further development of the theory. Stochastic partial differential equations have coefficients with very singular (Brownian) behavior. In the linear context, this can usually be handled by known methods, such as the classical martingale approach. This method is based on the linear character of the higher order part of the equation and thus cannot be used for nonlinear problems, where it is necessary to find appropriate alternative notions of solutions. These, in the context of first- and second-order nonlinear equations, are the stochastic viscosity and pathwise entropy solutions that have been introduced by the principal investigator and his collaborators. A part of the project is the study of the qualitative behavior/properties of these solutions. In the context of mathematical biology, the principal investigator plans to work on models of adaptation/selection as well as on models of the biology of development. The former concerns questions related to the adaptation of species to global change, the resistance of insects to pesticides, etc. The latter aims at developing models to study how positional information is provided to proliferating cells, the main questions being the formation and location of sharp and precise boundaries.