Mass Transfer and Evolution Problems, Free Boundary Problems

传质和进化问题、自由边界问题

• 批准号: 9970577
• 负责人: Mikhail Feldman
• 金额: $ 7.23万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970577&HistoricalAwards=false
• 关键词: Mass; Transfer; Evolution; Problems; Free

DMS-9970577ABSTRACTM. Feldman will work on three main topics. (1) Study of Monge-Kantorovichmass transfer problem in nonhomogeneous media, which include transporton manifolds with obstacles, and in anisotropic and nonsmooth spaces.Also, related evolution problems will be studied. Recently a progress was made by many authors in studying mass transport in the Euclidean space. Nonhomogeneous media is a natural next step. (2) Study of existence ofconvex viscosity solutions of geometric evolution equations, including nonlocal curvature motion, in the spatial dimensions three and higher. Such geometric evolutions are related to Monge-Kantorovich mass transfer. (3) Free boundary problems: existence and stability of solutions, and regularity of free boundaries.Methods of mass transfer theory were used recently in several applications including mathematical models in economics, meteorology, fluid dynamics, sandpiles growth and collapse, non-Newtonian fluids, plasticity. Free boundaries and moving interfaces arise naturally in many models.Mathematical analysis of such problems improves our understanding of qualitative properties of models, and is a necessary ingredient in developing of fast and reliable numerical algorithms. In addition, mathematical study sometimes discovers unexpected connections between different models (for example, partial differential equations and transport problems), and such interplay is useful in applications.

DMS-9970577ABSTRACTM。费尔德曼将研究三个主要主题。 （1）对非均匀培养基中的Monge-Kantorovichmass转移问题的研究，其中包括具有障碍物的运输歧管，以及各向异性和非平滑空间中的。最近，许多作者在研究欧几里得领域的大规模运输方面取得了进展。非官方媒体是自然的下一步。 （2）研究几何进化方程（包括非局部曲率运动）在三个及以上的空间尺寸中的几何进化方程的存在。这种几何发展与Monge-Kantorovich传质有关。 （3）自由边界问题：解决方案的存在和稳定性以及自由边界的规律性。最近在几种应用中使用了传质理论的方法，包括经济学，气象学，流体动力学，砂台球的生长和崩溃，非牛顿式流体，可塑性的数学模型。自由边界和移动界面自然出现在许多模型中。对此类问题的数学分析可以提高我们对模型定性特性的理解，并且是开发快速可靠的数值算法的必要组成部分。此外，数学研究有时会发现不同模型之间的意外连接（例如，部分微分方程和传输问题），并且这种相互作用在应用中很有用。