CAREER: Problems in regularity theory for linear and nonlinear partial differential equations

线性和非线性偏微分方程出现在许多数学领域，如微分几何、随机控制理论和数学金融。本项目主要研究粗糙系数的椭圆型和抛物型方程，线性层叠板和复合介质产生的系统，以及流体力学中的演化方程。项目的第一部分是系统地研究椭圆型和抛物型线性和非线性可能非定域方程在Sobolev空间中的正则性和强可解性。这项研究的新颖之处在于，在一些自变量中，系数只能被测量。该项目的第二部分涉及线性层压板和复合介质产生的偏微分系统。重点研究了系统弱解梯度的局部光滑性和全局光滑性以及边界点附近的高正则性。第三部分讨论了数学流体力学中几种非线性抛物方程模型的正则性理论。讨论了Navier-Stokes方程弱解的部分正则性和正则性准则的几个问题。拟议的研究将在生物学、物理学、经济学和金融学等不同领域有重要的应用。例如，描述流体流动的方程已被广泛用于模拟洋流、天气和气候、飞机周围的空气流动以及星系内恒星的运动，等等。它们有各种重要的应用，如气动形状的优化设计。为了传播他的工作并增加其影响，私家侦探将把这些研究项目整合到布朗大学本科生和研究生培训的更大框架中。除了招募和指导科学和工程专业的学生，特别强调代表性不足的群体，训练学生进行微分方程的研究外，拟议的教育活动还将包括组织一个关于流体力学方程的夏季讲习班，并指导本科生数学竞赛。

Linear and nonlinear partial differential equations arise in many areas of mathematics, such as differential geometry, stochastic control theory and mathematical finance. This project focuses on the investigations of elliptic and parabolic equations with rough coefficients, systems arising from linear laminates and composite media, and evolutionary equations from fluid mechanics. The first part of the project is to systematically investigate the regularity and the strong solvability in Sobolev spaces for both linear and nonlinear possibly nonlocal equations of elliptic and parabolic type. The novelty of this research is that coefficients are allowed to be merely measurable in some of the independent variables. The second part of the project concerns partial differential systems arising from linear laminates and composite media. Theemphases are in local and global smoothness of the gradient of weak solutions to these systems and the higher regularity near boundary points. The third part regards the regularity theory for several models of nonlinear parabolic equations in mathematical fluid mechanics. Several problems about the partial regularity as well as regularity criteria of weak solutions to the Navier-Stokes equations will be addressed.The proposed research will have significant applications in areas as diverse as biology, physics, economics, and finance. For instance, the equations describing fluid flow have been widely used to model ocean currents, the weather and climate, air flow around a airplane, and motion of stars inside a galaxy, to name a few. They have various important applications such as the optimal design of aerodynamic shapes. To disseminate his work and increase its impact, the P.I. will integrate these research projects into the larger framework of the undergraduate and graduate training at Brown University. Besides recruiting and advising students in science and engineering with a special emphasis on under-represented groups, training students in conducting research in differential equations, the proposed educational activities will alsoinclude the organization of a summer workshop on equations in fluid mechanics and mentoring undergraduate mathematical competitions.