Nonlinear Diffusion Equations and Free-Boundary Problems

这个项目主要研究非线性椭圆型和抛物型方程和自由边界问题，涉及更复杂的微分几何问题，包括高斯曲率流、Ricci流和具有非负高斯曲率的Weil问题，以及物理应用，如薄膜动力学和火焰传播。本项目的第一部分将研究由拟线性和完全非线性几何流动的退化引起的自由边界问题的几何和正则性。例如平边的高斯曲率流或包括调和流在内的更一般的曲率流。理解这种方程模型和自由边界问题可能具有重要的几何甚至拓扑应用。另一条新的研究方向将研究退化的Monge-Ampere方程和相关椭圆自由边界问题解的正则性。它的主要目的是发展新的技术来建立完全非线性退化椭圆型方程的最优正则性。第三部分的目的是研究Stefan类型自由边界问题中几何和正则性之间的联系以及奇点的形成，包括Hele-Shaw流动和火焰传播中的自由边界问题。在所提出的方法中，利用问题的几何方面是至关重要的。活动的最后一部分将研究各种奇异扩散模型解的渐近行为。特别地，它将讨论二维Ricci流极大解的第二类爆破行为。这些解对应于非紧曲面上的完备黎曼构形度量。该项目将广泛的数学活跃领域联系在一起，特别是非线性偏微分方程、几何和经典分析。所提出的关于退化非线性抛物方程和自由边界问题的几何和正则性的研究活动可能会产生重要的几何甚至拓扑应用。关于Stefan类型自由边界问题的研究活动与各种重要的物理模型密切相关，包括预混等扩散火焰在高活化能极限下的传播。本项目将研究的奇异扩散模型出现在各种物理应用中，如布居动力学、气体动力学和薄膜动力学。学生和博士后将作为本项目的一部分进行培训。将特别重视鼓励有才华的女性本科生、研究生和博士后在数学或相关科学领域取得成功。将为研究生设计和实施连接偏微分方程式和几何分析的新课程。

Title: Nonlinear Diffusion Equations and Free boundary problemsPI: Panagiota Daskalopoulos, Columbia UniversityABSTRACTThis project concerns with the study of nonlinear elliptic and parabolic equations and free-boundary problems, in connection with more complex problems of differential geometry,includingthe Gauss curvature flow, the Ricci flow and the WeylProblem withnonnegative Gaussian curvature and with physical applications such as thin liquidfilm dynamics and flame propagation.The first part of the project will study thegeometry and regularity of free-boundary problems arising from the degeneracy ofquasilinear and fully-nonlinear geometric flows,such as the Gauss curvature flow withflat sides or more general curvature flows including the Harmonic flow.The understanding of such models of equations and free-boundary problemsmay have significant geometric and even topological applications.A different new line of research will study the regularity of solutionsofdegenerate Monge-Ampere equations and related ellipticfree-boundary problems. Its main goal is to develop new techniques toestablish the optimal regularity in fully-nonlineardegenerate elliptic equations. The proposed work is also motivated bythe well known Weyl problem with nonnegative Gaussian curvature.The aim of the third part of the project is to study the connectionbetween the geometry and the regularity as well as the formation ofsingularities in Stefan type free-boundary problems including also theHele-Shaw flow and free-boundary problem in flame propagation. The useof the geometric aspects of the problems is crucial in the proposedapproach. The last part of the proposed activity will study the asymptoticbehavior of solutions of variousmodels of singular diffusion. In particular, it will deal with the type II blow up behavior of maximal solutions of the two dimensionalRicci Flow. These solutions correspond to complete Riemannian conformalmetrics on a non-compact surface.This project links a wide range of active fields of mathematics, inparticular nonlinear partial differential equations, geometry andclassical analysis.The proposed research activity on the geometry and regularity ofdegenerate nonlinear parabolic equations and free-boundary problems mayresult to significant geometric and even topological applications. Theproposed research activity on Stefan type free-boundary problems isclosely relatedto various important physical models, including the propagation of thepremixed equi-diffusional flames in the limit of high activation energy.The models of singular diffusion which will be studiedin this project arise in variousphysical applications such as population dynamics, the kinetic theory ofgases and thin liquid film dynamics.Students and postdocs will be trained as part of this project. Special emphasis will be given to theencouragement of talented femaleundergraduate students, graduate students and postdocsto pursue a successfulcareer in mathematics or related sciences. New courses linking PartialDifferential Equations and Geometric Analysis for graduate students will be designedand implemented.