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。该项目的重点是研究椭圆和抛物方程与粗糙系数，系统所产生的线性层压板和复合介质，和进化方程从流体力学。该项目的第一部分是系统地研究Sobolev空间中线性和非线性椭圆和抛物型可能非局部方程的正则性和强可解性。这项研究的新奇在于，允许系数在一些自变量中仅仅是可测量的。该项目的第二部分涉及线性层压板和复合介质所产生的偏微分系统。重点研究了这类方程组弱解梯度的局部光滑性和全局光滑性，以及在边界点附近的高正则性。第三部分是数学流体力学中几种非线性抛物型方程模型的正则性理论。本文将讨论Navier-Stokes方程弱解的部分正则性和正则性准则等问题，这些研究在生物学、物理学、经济学和金融学等领域都有重要的应用价值。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、飞机周围的气流以及星系内恒星的运动等。它们具有各种重要的应用，例如空气动力学形状的优化设计。为了传播他的工作并增加其影响，P.I.将这些研究项目纳入布朗大学本科生和研究生培训的更大框架。除了招收科学和工程专业的学生并为他们提供咨询意见，特别重视代表性不足的群体，培训学生进行微分方程研究，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，并指导大学生数学竞赛。

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.