Some elliptic and hyperbolic problems arising in mechanics

力学中出现的一些椭圆和双曲问题

• 批准号: 0908663
• 负责人: Daniel Spirn
• 金额: $ 10.04万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908663&HistoricalAwards=false
• 关键词: Some; elliptic; hyperbolic; problems; arising

ChenDMS-0908663 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigatorstudies some mathematical models that occur in three specificareas: electromagnetism, solid mechanics, and fluid mechanics. In the first project on electromagnetism, the investigator andhis collaborators establish an identification of the self-dualChern-Simons vortices, as well as study the electric and magneticeffects on the stability of the dually charged vortices. Thesecond project concerns the dynamics of gel swelling. Themathematical construction of well-posedness of classical and weaksolutions is proven in the context of hyperbolic conservationlaws. The third project involves the well-posedness andstability of various nonlinear dispersive equations arising fromsolid and fluid mechanics. Methods of mathematical analysis arethe primary tool employed in the investigations. Physical phenomena are usually well modeled via nonlinearpartial differential equations. Such equations are exceedinglydifficult to study, both theoretically and numerically, yet theirunderstanding is important to further progress of many areas ofphysics and engineering. One of the objectives of this projectis to study the behavior of electrically and magnetically chargedparticles in the classical field theory. Another objectiveconcerns the study of soft condensed matter, which is relevant todrug manufacturing and bacterial motility. The third objectiveis to understand a new type of transonic wave arising in solidmechanics, which is different from the context of transonic flowin gas dynamics. The fourth objective is to study water wavesthat may occur in the ocean that are affected by the earth'srotation, and to understand how they can form tsunamis and howthey can become turbulent. The results of this projectcontribute to the identification of physical problems of greatscientific importance that offer new opportunities for theintegration of applied analysis in research and in the trainingof graduate and undergraduate students.

ChenDMS-0908663 该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。 研究人员研究了三个特定领域中出现的一些数学模型：电磁学、固体力学和流体力学。在第一个电磁学项目中，研究者和他的合作者建立了自对偶陈-西蒙斯涡旋的识别，并研究了电磁效应对双电荷涡旋稳定性的影响。 第二个项目涉及凝胶膨胀的动力学。 经典解和弱解适定性的数学构造在双曲守恒定律的背景下得到了证明。 第三个项目涉及固体和流体力学产生的各种非线性色散方程的适定性和稳定性。 数学分析方法是调查中使用的主要工具。 物理现象通常可以通过非线性偏微分方程很好地建模。 这些方程在理论上和数值上都非常难以研究，但它们的理解对于物理学和工程学许多领域的进一步进步非常重要。 该项目的目标之一是研究经典场论中带电和磁性带电粒子的行为。 另一个目标涉及软凝聚态物质的研究，这与药物制造和细菌运动有关。 第三个目标是了解固体力学中出现的新型跨音速波，它不同于气体动力学中的跨音速流动的背景。 第四个目标是研究海洋中可能出现的受地球自转影响的水波，并了解它们如何形成海啸以及如何变得湍流。 该项目的结果有助于确定具有重大科学意义的物理问题，为应用分析在研究以及研究生和本科生培训中的整合提供新的机会。