Some elliptic and hyperbolic problems arising in mechanics

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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.