Nonlinear Parabolic Equations: Free-Boundary Problems and Singular Behavior

这项建议的主要目的是研究与退化和奇异扩散有关的非线性抛物型方程，结合更复杂的微分几何问题，包括高斯曲率流和Ricci流，以及物理应用，如多孔介质中的扩散和薄膜动力学。将研究的一个特定领域涉及简并扩散中界面的正则性和几何：这个长期项目始于1997年，目的是确定简并扩散中界面的几何和正则性之间的联系。另一个主要目标的具体领域涉及研究快速和超扩散的非线性抛物问题：这个项目的目标是研究几个新的重要现象，这些现象与物理应用和微分几何中出现的奇异扩散方程的解的适定性和消失轮廓有关。特别是，它涉及对Ricci流的最大解的消失轮廓和这些结果的几何含义的研究。在这个建议下要研究的非线性方程形成了许多被认为对技术和整个社会都很重要的应用的基本概念。从化学品到石油甚至水，材料的净化通常是通过过滤器的扩散来实现的。净化过滤器是建议书中描述的多孔介质。薄膜动力学和薄膜间的范德华作用力可用超快扩散的奇异拟线性方程来描述。种群增长、高分子链增长的动力学，包括生物分子的交联化和高速增长，也是适合我们基础研究的非线性现象。宇宙膨胀和其他宇宙学现象的有趣问题似乎是由非线性动力学支配的，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.