Nonlinear Degenerate Parabolic Problems and Related Topics

非线性简并抛物线问题及相关主题

• 批准号: 9801304
• 负责人: Panagiota Daskalopoulos
• 金额: $ 8.9万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801304&HistoricalAwards=false
• 关键词: Nonlinear; Degenerate; Parabolic; Problems; Related

DMS-9801304 P. Daskalopoulos The abstact follows : The major objectives of this proposal are concerned with the study of certain quasilinear or fully-nonlinear degenerate parabolic equations in connection with more complexproblems of differential geometry, including the Ricci flow and the Gauss curvature flow. These problems find physical applications in such areas as population dynamics, diffusion in porous media, and thin liquid film dynamics. One specific area which will be investigated is the regularity question in free-boundary problems arising from the degeneracy of certain non-linear parabolic equations, including the porous medium equation, the evolution p-laplacian equation and Gauss Curvature flow with flat sides. Another specific area of the main objectives concerns the study of ultra-fast diffusion parabolic equations, which have received large attention because of their relationship to differential geometry topics, such as the Ricci flow and the Yamabe flow. The solvability of the Cauchy problem for these equations, the study of the blow up profile of solutions, the nonradial structure solutions, and the uniqueness of solutions, are among the problems which will be investigated in this specific area. The still open question of the uniqueness of sign-solutions of the porous medium equation, in connection with the continuity question for a class of linear singular parabolic equations falls under the proposed major goals to be studied. 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 ultra-fast d iffusion. 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 goverened by nonlinear dynamics, which in certain cases are applications of the more complex problems described here.

DMS-9801304 P.Daskalopoulos本文的主要目的是研究某些拟线性或完全非线性退化抛物型方程与更复杂的微分几何问题，包括Ricci流和Gauss曲率流。这些问题在人口动力学、多孔介质中的扩散和薄液膜动力学等领域都有物理应用。一个具体的研究领域是由某些非线性抛物型方程的退化引起的自由边界问题的正则性问题，包括多孔介质方程，发展的p-拉普拉斯方程和具有平坦边的高斯曲率流。另一个主要目标的具体领域涉及超快扩散抛物方程的研究，由于其与微分几何主题的关系，如Ricci流和Yamabe流，受到了极大的关注。这些方程的柯西问题的可解性，解的爆破轮廓的研究，非径向结构解，解的唯一性，都是这一特定领域将要研究的问题。与一类线性奇异抛物型方程的连续性问题有关的多孔介质方程符号解的唯一性问题仍然是悬而未决的问题，这是提出要研究的主要目标之一。在这一建议下要研究的非线性方程形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，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。薄膜动力学和薄层间的范德华力可用超快扩散的奇异拟线性方程描述。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控制的，在某些情况下，这是这里描述的更复杂问题的应用。