退化与奇异偏微分方程在几何和物理中的应用

This project is mainly devoted to the study of several degenerate nonlinear elliptic equations which are closely related to geometry, several complex variables and algebraic geometry, with a particular emphasis on the existence of solutions in large, regularity, well-posedness, asymptotics and blow-up analysis. More precisely, we plan to investigate: elliptic boundary value problems with infinite degeneracy, elliptic problems with totally characteristic degeneracy, singular Yamabe equation and estimates on the eigenvalues of degenerate elliptic operators, higher dimensional Trudinger problem and embededness problem in large, the classification of stable and finite Morse index solutions of nonlinear elliptic equations, the analysis of Chern-Simons system and the corresponding Toda system, the existence of solutions to Chern-Simons-Higgs system with multiple vortex points; On the other hand, besides by using the analysis methods of such partial differential equations to study those kinds of geometry problems, we also plan to study some applied models of this type degenerate PDEs in material sciences and fluid dynamics, including models of composite material and complex fluids, incompressible fluid equations with two phases in porous medium, as well as the liquid crystals equations, where we plan to establish the regularity, long time behavior, existence of strong solutions, blow-up mechanism and the dimension of singular sets of solutions to these equations.

本课题主要研究与几何学、多复变和代数几何等方向密切相关的几类退化非线性椭圆型方程，特别是研究大范围解的存在性、正则性、适定性、渐近性与blow-up分析，包括无穷阶退化的椭圆边值问题，全特征退化的椭圆边值问题，锥型奇异Yamabe方程和退化椭圆算子特征值估计，高维Trudinger问题及大范围嵌入问题，非线性椭圆方程的稳定解及有限Morse指标解的分类，Chern- Simons系统及相应的Toda系统的解的分析，非对偶Chern-Simons-Higgs系统的多点漩涡解的研究等问题；另外除了使用偏微分方程这类分析的工具来研究这些具有几何背景的问题之外，我们还将研究这类退化偏微分方程在材料科学和流体力学中的某些应用型模型，包括研究混合材料及复杂流体模型，多孔介质中的不可压两相流方程组以及液晶流体力学方程解的正则性、长时间性态，强解的存在性、Blow-up 机理以及奇异集的的维数分析等问题。

本重点项目组在几何、多复变和代数几何等方面具应用背景的退化型椭圆算子特征值的上下界估计及渐近性，有限阶和无穷阶退化的椭圆边值问题和抛物方程的初边值问题，全特征退化的椭圆边值问题，高维Trudinger问题及大范围嵌入问题，非线性椭圆方程的稳定解及有限Morse指标解的分类，具生物和医学背景的Chemotaxis型方程的初边值问题，Chern- Simons系统及相应的Toda系统的解的分析，非对偶Chern-Simons-Higgs系统的多点漩涡解的研究，分数阶Yamabe问题，以及在研究退化型偏微分方程在材料科学和流体力学中的某些应用型模型，包括研究混合材料及复杂流体模型，多孔介质中的不可压两相流方程组以及液晶流体力学方程解的正则性、长时间性态，强解的存在性、Blow-up 机理以及奇异集的的维数分析等问题上，均取得了系列的研究成果