This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The investigator studies some mathematical models that occur in three specific areas: electromagnetism, solid mechanics, and fluid mechanics. In the first project on electromagnetism, the investigator and his collaborators establish an identification of the self-dual Chern-Simons vortices, as well as study the electric and magnetic effects on the stability of the dually charged vortices. The second project concerns the dynamics of gel swelling. The mathematical construction of well-posedness of classical and weak solutions is proven in the context of hyperbolic conservation laws. The third project involves the well-posedness and stability of various nonlinear dispersive equations arising from solid and fluid mechanics. Methods of mathematical analysis are the primary tool employed in the investigations.
Physical phenomena are usually well modeled via nonlinear partial differential equations. Such equations are exceedingly difficult to study, both theoretically and numerically, yet their understanding is important to further progress of many areas of physics and engineering. One of the objectives of this project is to study the behavior of electrically and magnetically charged particles in the classical field theory. Another objective concerns the study of soft condensed matter, which is relevant to drug manufacturing and bacterial motility. The third objective is to understand a new type of transonic wave arising in solid mechanics, which is different from the context of transonic flow in gas dynamics. The fourth objective is to study water waves that may occur in the ocean that are affected by the earth's rotation, and to understand how they can form tsunamis and how they can become turbulent. The results of this project contribute to the identification of physical problems of great scientific importance that offer new opportunities for the integration of applied analysis in research and in the training of graduate and undergraduate students.
