Div-curl systems and Variational principles

该项目将支持研究人员对科学和工程中感兴趣的某些基本方程组的分析。我们的主要课题是空间中有界区域上的div-curl系统的边值问题。 电磁场的麦克斯韦方程组和流体力学中的一些问题都受这个系统和某些相关的边界条件的支配。关于这些系统及其解决方案，仍然存在许多悬而未决的问题。拟议的研究集中在如何 描述这些边值问题的数学性质和近似解。这些信息对于开发计算这些场或流的良好算法是必需的--并且对于装置和设备的数值建模应该具有广泛的适用性。从数学上讲，这些div-curl系统是超定方程组，只有当某些相容条件成立时才有解--并且除了标准边界条件外，通常还需要额外的数据。这些额外的条件包括自然的分析条件和一些微妙的条件，从几何考虑。他们有时是已知的专家对这些问题，但他们的声明以前只是含糊地说，因为精确的版本需要相当多的几何和分析细节。PI及其合作者最近的论文中仔细描述了它们。这个建议是为了支持这些结果在应用中出现的一些模型中的实施和进一步发展。拟议的研究将使用的解决方案的变分表征，以获得尖锐的结果，这些系统的有限能量的解决方案，在一些重要的几何情况。我们还打算调查的各种不同的几何配置的解决方案的数值逼近的好方法的发展。一个特别的兴趣将是某些模型的铁磁体和静电放电的解决方案的分析。

This project will support the investigator's analysis of certain basic systems of equations of interest in science and engineering. Our primary topic will be boundary value problems for div-curl system on bounded regions in space. Both Maxwell's equations for an electromagnetic field and some problems in fluid mechanics are governed by this system with certain associated boundary conditions. There still are many open questions about these systems and their solutions. The proposed research centers on how to describe the mathematical properties, and approximation, of solutions of these boundary value problems. This information is needed for the development of good algorithms for the computation of these fields or flows -- and should have wide applicability for the numerical modeling of devices and equipment.Mathematically, these div-curl systems are over-determined systems of equations that only have solutions when certain compatibility conditions hold -- and often require extra data in addition to standard boundary conditions. These extra conditions include both natural analytical conditions and some subtle conditions that arise from geometrical considerations. They sometimes are known to experts on these subjects, but their statements have previously only been vaguely stated, since the precise versions require considerable geometrical and analytical detail. They have been carefully described in recent papers of the PI and his collaborators. This proposal is to support the implementation, and further development, of these results in some models that arise in applications. The proposed research will use the variational characterization of the solutions to obtain sharp results about the finite energy solutions of these systems in a number of important geometrical situations. We also intend to investigate the development of good methods for the numerical approximation of the solutions in various different geometrical configurations. A particular interest will be the analysis of the solution of certain models of ferromagnetic bodies and electrostatic discharges.