Aspects of regularity theory for PDE

该项目解决了偏微分方程理论中的几个公开问题。这些问题主要出现在流体力学中使用的方程（Navier-Stokes方程和Euler方程）或变分法中（向量值函数的多维变分积分极小值的正则性）。对于Navier-Stokes方程，研究将包括以下重点领域：（1）稳态解的长距离行为;（2）特殊类解的正则性，如轴对称解;和（3）具有低正则系数的相关线性方程的正则性。在欧拉方程的情况下，该项目将侧重于以下主题：（1）周期和准周期解的存在性;（2）二维方程稳态集的结构;（3）二维方程的各种稳态与二维统计理论的相关性。变分法中的问题涉及奇点的稳定性。与标量情形不同，向量值函数的正则变分泛函的极小元可以有奇点。这些奇点有多稳定？这个问题将得到解决。这个主题也与非线性弹性有关。偏微分方程的理论研究最终有一个非常实际的目标，那就是理解和预测方程解的行为。例如，流体流动通常由Navier-Stokes方程描述。这些方程的解是否正确地描述了我们在“实际”中看到的情况，我们能否利用这些方程对流动进行精确的预测？从我们对这些方程的了解来看，答案似乎是肯定的，但方程的数学仍然没有得到很好的理解。事实上，我们还没有令人满意的数学解释的行为，可以观察到的解决方案，无论是在实验中或在计算机模拟。我们需要从计算中获得的信息通常可以用简单而具体的术语来表达。例如，飞机在什么速度下会失速？这个问题很简单，然而这个问题背后的解的行为却很复杂。我们能否通过找到仍然“可控”的流的最重要的数学参数来管理复杂性？当我们研究解的正则性时，情况非常相似：似乎有一个非常大的解可以表现出的行为谱。我们能否通过关注相对较少的“正确”参数来获得对解的某种控制？什么是正确的参数？这个研究项目的重点是这种性质的问题。如果我们能够确定一些控制解的行为并且我们能够控制的好的量，那么计算解本身就会变得容易得多，因为理论信息告诉我们应该把计算资源集中在哪里。

The project addresses several open problems in the theory of partial differential equations. The problems arise primarily in the context of equations used in fluid mechanics (the Navier-Stokes equations and Euler's equations) or in the calculus of variations (regularity of minimizers of multi-dimensional variational integrals for vector-valued functions). For the Navier-Stokes equations the study will include the following areas of emphasis: (1) long-distance behavior of steady-state solutions; (2) regularity for special classes of solutions, such as the axi-symmetric solutions; and (3) regularity of related linear equations with low-regularity coefficients. In the case of Euler's equations, the project will focus on the following topics: (1) existence of periodic and quasi-periodic solutions; (2) the structure of the set of the steady-states of the two-dimensional equations; and (3) the relevance of various steady-states of the two-dimensional equations for two-dimensional statistical theories. The problems in the calculus of variations concern the stability of singularities. Unlike in the scalar case, the minimizers of regular variational functionals for vector-valued functions can have singularities. How stable are these singularities? This question will be addressed. The topic also has connections to nonlinear elasticity. Theoretical research in partial differential equations ultimately has a very practical goal, which is the understanding and prediction of behavior of solutions of the equations. For example, fluid flows are usually described by the Navier-Stokes equations. Do the solutions of these equations correctly describe what we see "in practice," and can we use the equations to make precise predictions about the flows? From what we know about these equations, the answers seem to be yes, but the mathematics of the equations is still not very well understood. Indeed, we do not yet have satisfactory mathematical explanations for the behavior of solutions that can be observed either in experiments or in computer simulations. The information we need to obtain from the computations can usually be formulated in simple and concrete terms. For instance, at what speed will an aircraft stall? The question is simple, whereas the behavior of the solutions underlying this question is complicated. Can we somehow manage the complexity by finding the most important mathematical parameters of the flow that are still "controlable"? When we study regularity of solutions, the situation is quite similar: there seems to be a very large spectrum of behaviors that solutions can exhibit. Can we obtain some control of the solutions by focusing on relatively few "right" parameters? What are the right parameters? This research project focuses on questions of this nature. If we can identify some good quantities that govern the behavior of the solutions and that we are able to control, then it becomes much easier to calculate the solutions themselves, since the theoretical information tells us where to focus our computational resources.