Non-linear partial differential equations, free boundary problems and fractional operators

这个项目涉及非线性偏微分方程的数学分析的几个方面。大部分的努力致力于研究椭圆型和抛物型自由边界问题。主要研究者对这类问题的存在性和正则性理论感兴趣，并将特别注意研究解的渐近行为（如长时间行为和均匀化极限）。另一个研究方向涉及非局部椭圆型和抛物型方程，主要是涉及分数阶拉普拉斯算子的方程。特别是，该项目将研究这些微分算子的非局域性对众所周知的现象，如前传播和均匀化的影响。最后，对三阶和四阶非线性椭圆型和抛物型方程进行了研究。对这类方程的分析仍然知之甚少，主要是因为缺乏二阶方程所具有的极大值原理和其他基本性质。发展这些方程的存在唯一性理论将是该项目的主要挑战之一。非线性偏微分方程在物理学、生物学、经济学、工程学和其他科学领域都有应用。例如，自由边界问题通常出现在涉及（两种材料之间）随时间变化的界面的物理现象的建模中（一个很好的例子是研究一块冰的融化）。本文所讨论的自由边界问题在气体燃烧模拟和流体流动研究中经常出现。更好地理解介质性质（例如，冰中的杂质）的微小不均匀性的影响是前面提到的“均匀化理论”的目标。最终，这一理论将允许发展更精确的模型来描述各种各样的现象。非局部扩散方程是另一种通常用于模拟广泛现象的方程。首席研究员提出了一种新的方法，利用所谓的微观（动力学）模型来推导这些方程，其中的参数可以很容易地与简单的物理量联系起来。这将提高物理科学家使用的模型的准确性。最后，高阶椭圆型和抛物型方程的应用也很多。本文讨论的方程在水力裂缝的建模（例如，用于扩展油气储层中的岩石裂缝以提高采收率）和生物膜的研究中有一定的应用。

This project deals with several aspects of the mathematical analysis of nonlinear partial differential equations. A large part of the endeavor is devoted to the study of elliptic and parabolic free boundary problems. The principal investigator is interested in the existence and regularity theory for such problems, and special attention will be paid to investigating the asymptotic behavior of the solutions (such as the long-time behavior and homogenization limits). Another direction of research concerns nonlocal elliptic and parabolic equations, mainly equations involving fractional Laplace operators. In particular, the project will investigate the effects of the nonlocality of these differential operators on well-known phenomena such as front propagation and homogenization. Finally, part of this project is devoted to certain nonlinear elliptic and parabolic equations of third and fourth order. The analysis of such equations is still poorly understood, mainly because of the lack of a maximum principle and other basic properties that their second-order counterparts possess. Developing existence and uniqueness theories for such equations will be one of the main challenges of the project.Nonlinear partial differential equations have applications in physics, biology, economics, engineering, and other areas of science. Free boundary problems, for instance, typically arise in the modeling of physical phenomena that involve interfaces (between two materials) that are changing with time (a good example is the study of the melting of a block of ice). Some free boundary problems discussed in this proposal turn up in the modeling of gas combustion and in the study of fluid flow. A better understanding of the effects of small inhomogeneities in the properties of the medium (for instance, impurities in the ice) is the goal of the "homogenization theory" mentioned earlier. Ultimately, this theory will allow the development of more accurate models to describe a great variety of phenomena. Nonlocal diffusion equations are another type of equation that are commonly used to model a broad range of phenomenon. The principal investigator proposes a new approach to deriving these equations using so-called microscopic (kinetic) models, for which the parameters can be easily related to simple physical quantities. This will lead to better accuracy in the models used by physical scientists. Finally, applications of higher order elliptic and parabolic equations are also numerous. The equations discussed in this proposal have some applications to the modeling of hydraulic fractures (which are used, for example, to propagate rock fractures in oil and gas reservoirs so as to enhance oil recovery) and to the study of biological membranes.