Some elliptic and hyperbolic problems arising in mechanics

该奖项是根据2009年美国复苏和投资法案（公法111-5）资助的。研究者研究了出现在三个特定领域的一些数学模型：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一个自对偶的chen - simons涡的识别，并研究了对偶带电涡稳定性的电和磁效应。第二个项目涉及凝胶膨胀的动力学。在双曲守恒的背景下，证明了经典解和弱解的适定性的数学构造。第三个项目涉及由固体和流体力学引起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。通常用非线性偏微分方程可以很好地模拟物理现象。这些方程在理论上和数值上都很难研究，但对它们的理解对物理学和工程学的许多领域的进一步发展至关重要。本课题的目标之一是在经典场论中研究带电粒子和带电粒子的行为。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一种新型跨音速波，它不同于跨音速气体动力学的背景。第四个目标是研究受地球自转影响的海洋中可能出现的水波，并了解它们是如何形成海啸的，以及它们是如何变得湍流的。该项目的结果有助于识别具有重大科学意义的物理问题，为在研究和研究生和本科生的培训中整合应用分析提供了新的机会。

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.