The dynamics of a large number of `particles' can be described by the Boltzmann theory. Such `particles' can be as tiny as electrons, or as enormous as galaxies. The first part of the proposal is devoted to the mathematical study of stable equilibria in a collection of dilute electrons and ions, in the presence of electromagnetic interaction as well as inter-particle collisions. Such an investigation will base on a recent nonlinear energy method. Moreover, boundary effects will also be studied for such electrons and ions. Based on a recent variational approach, the second part of the proposal is devoted to the mathematical study of stable galaxy configurations in stellar dynamics, which is governed by the collisionless Boltzmann theory. The last part of the proposal is devoted to the mathematical study of dynamical instability of various important equilibria in many physical applications, such as Rayleigh-Taylor instability in fluid mechanics, Kruskal-Schwarzchild instability in MHD, as well as morphological instability of planar dissolution fronts in geology. Recent method of Strauss and the PI for nonlinear instability will play a crucial role in such a study.
A plasma is a collection of free moving, dilute charged particles. A plasma TV, a plane diode, as well as the hot air around a space shuttle, are all examples of plasma on earth. Although more than ninety five percent of matters in the universe are plasma, the center of plasma study is about the nuclear fusion. The first part of the proposal will lead to theoretical understanding of such important problems as ollissions in a plasma, control of plasma-wall interaction, as well as stable plasma configurations in a nuclear fusion device, e.g. a tokamak. The second part of the proposal will continue to search for physically reasonable and stable models for various galaxy configurations in our universe. The third part of the proposal will lead to a deeper understanding of various instabilities in physics. Such instabilities are believed to be the first stage of turbulence in fluids. Moreover, the stability study of dissolution fronts in rocks will eventually contribute to a better understanding of environmental change on earth over a long period of time.
