Mathematical Sciences: Nonlinear Partial Differential Equations

摘要Sverak 9622795 Sverak将研究非线性偏微分方程组的各种问题。其中包括：（1）研究向量值函数的一般多重变分积分，它们的欧拉-拉格朗日方程，密切相关的Morrey准凸性条件和其他椭圆性概念，以及与这些椭圆性条件相关的某些自然对象（半凸包）。这些调查应有助于了解一些非线性系统的整体性能的解决方案。 (2)三维不可压Navier-Stokes方程的Liouville型定理的研究及其在研究这些方程解的（潜在）奇异性中的应用。 研究变分积分的动机来自于一个非常普遍和非常重要的经典原理，即工程、物理和纯数学中的许多方程都可以直接与能量最小化（适当定义）联系起来。例如，描述材料在应力作用下变形的基本方程就与这一原理密切相关。虽然我们对数学的理解 最小化在许多重要的情况下是相当好的，但也有许多具有相当实际重要性的情况，我们的理解相当差。上面提到的材料变形方程给出了这种情况的一个很好的例子（除了材料中的应力相对较小的情况）。 我希望，对上述第一圈问题的研究将有助于解决围绕能量最小化的未决问题。事实证明，与这些问题相关的结果也可以帮助阐明其他数学问题，这些问题最初似乎与能量最小化没有直接关系（例如非线性振荡）。然而，更仔细的分析表明，有一些意想不到的联系。 对围绕能量最小化的数学问题的理解是 从实践的角度来看，这一点也很重要：如果我们在计算机上模拟最小化，那么我们所知道的非平凡数学事实可能会为我们节省大量的计算。事实上，在许多情况下，一个好的理论分析可以使一个攻击的问题，在第一次似乎遥不可及，甚至最快的计算机。人们可以提出类似的论点来解释上面提出的第二个问题背后的动机。事实上，在这种情况下，对良好的数学分析的需要可能更加明显：众所周知，在许多具有重要实际意义的情况下，即使在最快的计算机的帮助下，也没有可靠的方法来计算流体流动（由Navier-Stokes方程描述）。我希望，拟议的调查将增加我们对这些方程的解的行为的理解。

Abstract Sverak 9622795 Sverak's will study a variety of questions from system of non-linear partial differential equations. These include: (1) The study of general multiple variational integrals for vector valued functions, their Euler-Lagrange equations, the closely related Morrey's quasi-convexity condition and other notions of ellipticity, together with certain natural objects (semi-convex hulls) associated with these ellipticity conditions. These investigation should help to understand global properties of solutions of a number of non-linear systems. (2) The study of Liouville-type theorems for the three-dimensional incompressible Navier-Stokes equations and their applications to the study of (potential) singularities of solutions of these equations. The motivation for the study of variational integrals comes from the very general and extremely important classical principle that many equations arising in engineering, physics, and also pure mathematics can be directly linked to minimization of (suitably defined) energy. For example, the basic equations describing deformations of materials under stress are very closely related to this principle. Although our mathematical understanding of minimization is quite good in many important cases, there are also many situations of considerable practical importance where our understanding is rather poor. The above mentioned equations for deformations of materials give a good example of such situation (except for the case when the stress in the material is relatively small). I expect that the study of the first circle of problems proposed above will shed some light on open questions surrounding energy minimization. It turns out that results related to these questions can also help to elucidate other mathematical problems which at first do not seem to be directly related to energy minimization (such as non-linear oscillations). However, a closer analysis shows that there are some unexpected connections. The understanding of mathematic al problems surrounding energy minimization is also quite important from the practical point of view: if we simulate minimization on a computer, then non-trivial mathematical facts we know can potentially save us a considerable amount of computations we need to do. In fact, in many cases a good theoretical analysis can enable one to attack problems which at first seem out of reach of even the fastest computers. One can put forward similar arguments to explain the motivation behind the second circle of problems proposed above. In fact, in this case the need for good mathematical analysis is perhaps even more apparent: it is well known that in many cases of great practical importance, there is simplyno known method to reliably calculate fluid flows (describedby the Navier-Stokes equations), even with the help of the fastest computers. I hope that the proposed investigations will increase our understanding of the behavior of solutions of these equations.