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.

公司简介 该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。 电磁学研究电磁学、固体力学和流体力学三个领域的数学模型。在第一个关于电磁学的项目中，研究者和他的合作者建立了自对偶Chern-Simons涡旋的识别，并研究了电和磁对双电荷涡旋稳定性的影响。 第二个项目关注凝胶溶胀的动力学。 在双曲守恒律的背景下证明了经典解和弱解的适定性的数学构造。 第三个项目涉及固体和流体力学中各种非线性色散方程的适定性和稳定性。 数学分析方法是研究的主要工具。 物理现象通常可以通过非线性偏微分方程很好地建模。 这类方程无论从理论上还是数值上都很难研究，但对它们的理解对物理学和工程学许多领域的进一步发展都很重要。 本项目的目标之一是研究经典场论中带电和带磁粒子的行为。 另一个目标是研究软凝聚物质，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产生的一种新型跨音速波，它不同于气体动力学中的跨音速流动。 第四个目标是研究海洋中可能发生的受地球自转影响的水波，并了解它们如何形成海啸以及它们如何变得动荡。 该项目的结果有助于确定具有重要科学意义的物理问题，为研究和研究生和本科生的培训中应用分析的整合提供了新的机会。