Regularity Versus Singularity Formation in Nonlinear Partial Differential Equations

本项目从定性和定量的角度，关注与正则性和奇点形成二分法相关的偏微分方程理论的各个方面。作为项目的一部分，要研究的主题在数学领域是普遍的，因为同样的范式出现在例如几何，数学物理或动力系统中。该项目旨在理解的基本问题是，对于给定的偏微分方程系统，解的存在性和规律性与奇点的出现之间的相互作用。这类问题对于时变方程和时变方程都非常重要。本项目为研究生和博士后提供科研培训机会。本项目处于偏微分方程、几何测度理论、几何分析、谐波分析等多个数学领域的交叉点，在液晶、流体力学、统计物理、微分几何等各个领域的相关应用激发了许多问题的研究。首席研究员（PI）计划通过深入研究边界效应普遍存在的新模型，对最小曲面和自由边界谐波映射理论中出现的偏微分方程的正则性理论和几何性质进行研究。非局部方程理论使得研究流形边界上的连通和或具有自由边界的调和映射流的爆破解的构造等初看上去是局部的但本质上是非局部的问题成为可能。PI计划继续进行这一富有成果的研究。与后者相关的是，与分数阶拉普拉斯函数相关的一类特别重要的退化/奇异偏微分方程的理论最近有了一些发展。基于最近的成果，PI计划研究涉及变系数的流体动力学中的退化/奇异方程。可压缩流体的几个模型是这一研究方向的主要成果。几何微局部分析是理解系统奇异性和简并性的一种可能方法。PI和合作者多年来开发了一种有用的工具，称为抛物线粘合，它提供了一种非常通用的方法来构建新的物体，可能是单一的，在抛物线方程和几何流中。作为这个项目的一部分，PI计划继续发展这种方法来研究几个抛物方程中的冒泡现象，这些方程来自物理学（特别是流体动力学）和几何（例如曲率流和复杂流）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with various aspects of the theory of partial differential equations related to the dichotomy between regularity and singularity formation, from both a qualitative and a quantitative point of view. The topics to be studied as part of the project are universal in the field of mathematics in the sense that the same paradigm appears, for instance, in geometry, mathematical physics, or dynamical systems. The fundamental issue the project aims to understand is the interplay, for a given system of partial differential equations, between the existence and regularity of solutions versus the appearance of singularities. This type of question is of major importance for both time-dependent and time-independent equations. The project provides research training opportunities for graduate students and postdoctoral researchers.The project lies at the interface of several areas of mathematics, such as Partial Differential Equations, Geometric Measure Theory, Geometric Analysis, and Harmonic Analysis, with many problems under consideration being motivated by relevant applications to liquid crystals, fluid dynamics, statistical physics, and various areas of differential geometry. The Principal Investigator (PI) plans to pursue research in the regularity theory and geometric properties for partial differential equations arising in the theory of minimal surfaces and harmonic maps with free boundaries, by investigating thoroughly a new model where the boundary effects are prevalent. The theory of nonlocal equations makes it possible to study problems which seem to be local at first sight but are intrinsically nonlocal, such as connected sums on the boundary of manifolds or the construction of blow-up solutions of the harmonic map flow with free boundary. The PI plans to continue this fruitful line of research. Related to the latter, the theory of a particularly important class of degenerate/singular partial differential equations connected to the fractional Laplacian has seen some recent developments. Building on recent results, the PI plans to study degenerate/singular equations in fluid dynamics involving variable coefficients. Several models in compressible fluids are a major output of this line of investigation. A possible way to understand the singularities and degeneracies in a system is via geometric microlocal analysis. A useful tool developed over the years by the PI and collaborators, called parabolic gluing, offers a very versatile method to construct new objects, possibly singular, in parabolic equations and geometric flows. As part of this project, the PI plans to continue to develop this method to investigate bubbling phenomena in several parabolic equations coming from physics (fluid dynamics, in particular) and geometry (curvature flows and complex flows, for example).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.