Mathematical Sciences: Nonlinear Diffusion Equations and Related Elliptic Problems

摘要本课题研究了一类非线性扩散方程及相关的奇异椭圆型和线性退化抛物型问题。更确切地说，我们的工作可以分为以下几个项目：1.奇异抛物方程的快速和超快扩散(与M.Del Pino合作)。2.具奇异行为的半线性椭圆型方程(与M.Del Pino合作)。3.多孔介质方程符号解的唯一性--某些退化线性抛物型方程的连续性结果(与C.Kenig合作)。所提出的方程具有物理意义和理论意义。例如，从物理角度来看，它们是作为薄膜液膜动力学的模型出现的，也是作为服从玻尔兹曼方程的两种气体动力学中的极限密度分布的模型出现的。此外，在冶金和聚合物科学等其他领域，人们对这类方程的兴趣也由来已久。从数学的角度来看，它们表现出一种定性的行为，具有显著的特征，不同于以前研究的案例。他们的研究导致了数学分析新技术的发展。提出的问题的另一个吸引人的特点是它们与微分几何的联系，因为它们出现在所谓的Ricci流中，该流最近得到了显著的关注。

ABSTRACT This project involves the study of certain Nonlinear Diffusion Equations and related Singular Elliptic and Linear Degenerate Parabolic problems. More precisely, the proposed work can be divided into the following projects: 1. Singular Parabolic equations of Fast and Super-Fast diffusion (in collaboration with M. Del Pino). 2. Semilinear Elliptic equations with Singular behavior (in collaboration with M. Del Pino). 3. Uniqueness of signed solutions to the Porous Medium equation - A continuity result for certain Degenerate Linear Parabolic equations (in collaboration with C. Kenig). The equations proposed have both physical and theoretical significance. For example, from the physical point of view, they arise as models for the dynamics of thin liquid films and also as models for the limiting density distribution in the kinetics of two gases obeying Boltzmann equation. Also, interest in such equations has long existed in other fields such as metallurgy and polymer science. From the mathematical point of view, they exhibit a qualitative behavior that has remarkable features, different than in previously studied cases. Their investigation leads to the development of new techniques of mathematical analysis. Another attractive feature of the proposed problems is their connection to Differential Geometry, as they arise in the so called Ricci Flow, which has recently gained remarkable attention.