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系统的分析。我们的主要主题是空间中有界区域上div-curl系统的边值问题。电磁场的麦克斯韦方程和流体力学中的一些问题都是由这个系统和某些相关的边界条件所支配的。关于这些系统及其解决方案，仍有许多悬而未决的问题。所提出的研究集中在如何描述这些边值问题解的数学性质和逼近。这些信息对于开发计算这些场或流的良好算法是必需的，并且对于设备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，只有在某些兼容条件成立时才有解--并且除了标准边界条件之外，通常还需要额外的数据。这些额外的条件既包括自然的分析条件，也包括一些由于几何考虑而产生的微妙条件。它们有时为这些主题的专家所知，但它们的陈述以前只被含糊地陈述过，因为精确的版本需要相当多的几何和分析细节。在国际和平组织和他的合作者最近的论文中对它们进行了仔细的描述。这项提议是为了支持在应用中出现的一些模式中落实和进一步发展这些成果。所提出的研究将利用解的变分特征来获得关于这些系统在一些重要几何情况下的有限能量解的精确结果。我们还打算研究各种不同几何构型的解的数值逼近的良好方法的发展。特别感兴趣的是对某些铁磁体和静电放电模型的解的分析。

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.