Nonlinear Parabolic Equations: Free-Boundary Problems and Singular Behavior

非线性抛物型方程：自由边界问题和奇异行为

• 批准号: 0312006
• 负责人: Panagiota Daskalopoulos
• 金额: $ 6.95万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0312006&HistoricalAwards=false
• 关键词: Nonlinear; Parabolic; Equations; Free; Boundary

The major objectives of this proposal are concerned with the study of nonlinear parabolic equations related to degenerate and singular diffusion, in connection with more complex problems of differential geometry, including the Gauss curvature flow and the Ricci flowand with physical applications such as diffusion in porous media and thin liquid film dynamics.One specific area which will be investigated concerns with the regularity and geometry of interfaces in degenerate diffusion: this long term project which was initiated in 1997, has the objective of determining the connection between the geometry and regularity of the interfaces in degenerate diffusion. Another specific area of the main objectives concerns the study fast and super-diffusive nonlinear parabolic problems: this project has the objective of studying several new important phenomena related to the well-posedness and vanishing profile of solutions to singular diffusion equations arising in physical applications and differential geometry. In particular,it involves the study of the vanishing profile of the maximal solutions to the Ricci flow and the geometric implications of these results.The non-linear equations to be studied under this proposal form the basic concepts of many applications which deem to be important to technology and the society at large. The purification of materials, from chemicals to petroleum and even water, is often achieved by diffusion through filters. The purification filters are the porous media described in the proposal. Thin film dynamics and the Van der Waals forces operating between thin layers are described by singular quasilinear equations of super-fast diffusion. The dynamics of population growth, polymer chain growth, including cross linking and high rate growth of biomolecules, are also non-linear phenomena amiable to our basic studies. The interesting problem of the expanding universe and other cosmological phenomena seem to be governed by nonlinear dynamics, which in certain cases are applications of the more complex problems described here.

本建议的主要目标是研究与退化和奇异扩散有关的非线性抛物型方程，以及更复杂的 微分几何，包括高斯曲率流和里奇流以及 物理应用 如多孔介质中的扩散和薄液膜动力学。其中一个具体的领域，将被调查的关系，在退化扩散的界面的规则性和几何形状：这个长期的项目，这是在1997年发起的，有确定的几何形状和退化扩散中的界面的规则性之间的连接的目标。主要目标的另一个特定领域涉及研究快速和超扩散非线性抛物问题：该项目的目标是研究 在物理应用和微分几何中出现的奇异扩散方程解的适定性和消失轮廓有关的几个新的重要现象。特别是，它涉及到的Ricci流的最大解的消失轮廓和这些结果的几何含义的研究。在这个建议下，非线性方程的研究形成了许多应用的基本概念，认为是重要的技术和社会。从化学品到石油甚至水的材料的净化通常通过过滤器的扩散来实现。净化过滤器为建议书中描述的多孔介质。薄膜动力学和货车范德华力的薄层之间的操作描述的奇异准线性方程的超快扩散。种群增长、高分子链增长，包括生物分子的交联和高速增长的动力学，也是我们基础研究的非线性现象。膨胀宇宙和其他宇宙学现象的有趣问题似乎是由非线性动力学控制的，在某些情况下，这是这里描述的更复杂问题的应用。