PDE Methods for the Stability Study in Kinetic Theory and Their Applications

摘要：玻尔兹曼理论可以描述大量“粒子”的动力学特性，而玻尔兹曼理论可以描述大量“粒子”的动力学。这些“粒子”可以像电子一样小，也可以像星系一样大。该提案的第一部分致力于在电磁相互作用和粒子间碰撞存在的情况下，稀电子和离子集合中稳定平衡的数学研究。这样的研究将基于最近的非线性能量方法。此外，还将研究这类电子和离子的边界效应。基于最近的变分方法，提案的第二部分致力于恒星动力学中稳定星系结构的数学研究，这是由无碰撞玻尔兹曼理论控制的。论文的最后一部分致力于在许多物理应用中对各种重要平衡的动力学不稳定性进行数学研究，如流体力学中的瑞利-泰勒不稳定性，MHD中的克鲁斯卡尔-施瓦西不稳定性，以及地质学中平面溶解前沿的形态不稳定性。最近的Strauss方法和PI法对非线性不稳定性的研究将起到至关重要的作用。等离子体是自由运动的稀带电粒子的集合。等离子电视、平面二极管以及航天飞机周围的热空气都是地球上等离子体的例子。虽然宇宙中95%以上的物质是等离子体，但等离子体研究的中心是核聚变。该提案的第一部分将从理论上理解等离子体中的碰撞、等离子体-壁相互作用的控制以及核聚变装置（例如托卡马克）中稳定的等离子体配置等重要问题。提案的第二部分将继续为我们宇宙中的各种星系结构寻找物理上合理和稳定的模型。提案的第三部分将导致对物理学中各种不稳定性的更深入的理解。这种不稳定性被认为是流体湍流的第一阶段。此外，岩石溶解锋的稳定性研究将最终有助于更好地理解地球上长期的环境变化。

Abstract: award DMS-0305161, Yan Guo, Brown UniversityTitle: PDE methods for stability study in kinetic theory and their applicationsAbstract:The dynamics of a large number of `particles' can bedescribed 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.