Some elliptic and hyperbolic problems arising in mechanics

CHEN DMS-0908663该奖项是根据2009年《美国复苏和再投资法案》(公法第111-5条)资助的。研究人员研究发生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。在第一个关于电磁学的项目中，研究人员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。第二个项目涉及凝胶膨胀的动力学。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。数学分析方法是调查中使用的主要工具。物理现象通常通过非线性偏微分方程来很好地模拟。这样的方程无论在理论上还是在数值上都非常难研究，但对它们的理解对许多物理和工程领域的进一步发展是重要的。这个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，它与药物制造和细菌运动有关。第三个目的是理解固体力学中出现的一种不同于气体动力学中跨音速流动的新型跨音速波。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，以及它们如何变得湍急。该项目的结果有助于确定具有重大科学意义的物理问题，为将应用分析整合到研究和研究生和本科生的培训中提供了新的机会。

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.