Problems in Regularity Theory for Nonlinear Partial Differential Equations

本文的目的是研究各种非线性偏微分方程解的正则性。它们包括：(I)作为欧拉-拉格朗日方程出现的拟线性椭圆组，以及变分中的一般重积分和守恒律的双曲型方程组。我们将从补偿紧性理论的角度来研究这些方程解的正则性研究中出现的问题，这为研究这些看似多样的问题提供了一个统一的平台。我们期望Stefan Muller和PI最近发展的关于补偿紧性的新方法将使我们能够回答一些相对古老的开放问题。(Ii)Navier-Stokes方程。在这里，我们计划研究重要的特殊解类的内部正则性和边界正则性。这些研究的动机如下：由于通解的正则性问题是出了名的难解，所以研究一些非平凡的特殊情况将是有用的。我们还计划研究2×2矩阵上函数的Morrey拟凸性和局部椭圆性条件之间的关系。这个问题最近已经与调和分析和拟共形映射理论的一些古老而困难的猜想联系在一起，我们期望这些不同领域的思想的相互作用将是卓有成效的。这个建议的主要目的是研究非线性偏微分方程解的正则性。这样的方程出现在许多真实世界过程的模型中。数学家的任务就是解这些方程式。这通常是通过计算机上的数值模拟来完成的。为了能够以可靠的方式进行这样的模拟，必须对解的定性行为有所了解。例如，如果我们期望解在很小的距离内剧烈振荡，我们可能需要一种不同的方法，而不是在解仅缓慢变化的情况下。粗略地说，正则性理论的主要任务之一是发展方法，使我们能够可靠地预测解的行为，以便我们能够适当地选择求解方程的方法。拟议中的研究有望提高我们对这些困难和重要问题的理解。

The proposal is aimed at the study of regularity properties of solutionsof various systems of nonlinear partial differential equations. These include:(i) Quasilinear elliptic systems arising as Euler-Lagrange equationsof general multiple integrals in the Calculus of Variations and hyperbolic systems of conservation laws. Questions arising in the study ofregularity of solutions of these equations will be studied from the pointof view of the theory of Compensated Compactness, which provides a unifyingplatform for approaching these seemingly diverse problems. We expect that new methods in Compensated Compactness recently developedby Stefan Muller and the PI will enable us to give answers to some relativelyold open questions.(ii) The Navier-Stokes equations. Here we plan to study both the interiorand boundary regularity for important special classes of solutions.The motivation for these investigation is the following: since regularityquestions for general solutions are notoriously intractable, it will be usefulto study some non-trivial special cases. Hopefully this will provide insights for the general case.We also plan to study the relationship between Morrey's quasiconvexityand local ellipticity conditions for functions on two by two matrices.This problem has recently been linked to some old and difficult conjecturesin Harmonic Analysis and the theory of quasi-conformal mappings, and we expectthat the interaction of ideas from these different areas will be fruitful.The main aim of this proposal is the study of regularity properties ofsolution of nonlinear partial differential equations. Such equationsappear in many models of the real-world processes. The task ofmathematicians is to solve these equations. Often thisis done by a numerical simulation on a computer. To be able to dosuch simulations in a reliable way, one must know something aboutthe qualitative behavior of the solutions. For example, if we expect thatthe solutions will oscillate wildly over small distances, we might needa different method than in the case when the solutions change onlyslowly. Roughly speaking, one of the main task of regularitytheory is to develop methods which would enable us to reliablypredict the behavior of the solutions, so that we can then chooseadequately the methods for solving the equations. The proposed research will hopefully improve our understandingof these difficult and important questions.