Nonlinear Parabolic Equations: Free-Boundary Problems and Singular Behavior

本提案的主要目标是研究与退化和奇异扩散有关的非线性抛物方程，与更复杂的微分几何问题有关，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。将研究的一个特定领域涉及简并扩散中界面的规律性和几何形状：这个长期项目始于1997年，其目标是确定简并扩散中界面的几何形状和规律性之间的联系。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与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剖面相关的几个新的重要现象。特别地，它涉及到里奇流的最大解的消失轮廓和这些结果的几何意义的研究。在这个提议下要研究的非线性方程构成了许多应用的基本概念，这些应用被认为对技术和整个社会都很重要。从化学品到石油，甚至水的净化，通常是通过过滤器扩散来实现的。净化过滤器为提案中描述的多孔介质。用超高速扩散的奇异拟线性方程描述了薄膜动力学和作用于薄层之间的范德华力。人口增长的动态，聚合物链的增长，包括交联和生物分子的高速增长，也是非线性现象，有利于我们的基础研究。宇宙膨胀的有趣问题和其他宇宙现象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。

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.