Dynamics of Gaseous Stars and Hydrodynamic Limits for Boltzmann Equations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目的第一个目标是更好地理解物理真空作为可压缩欧拉和纳维-斯托克斯方程解的自由边界，有或没有强迫，并发展一个令人满意的非线性稳定性理论。物理真空自然出现在天体物理学和气体动力学中：对其行为的数学研究将有助于理解气态恒星的动力学。此外，这项研究将导致退化双曲方程的一般理论。第二个目标是通过严格的论证，从玻尔兹曼方程推导出各种流体方程。玻尔兹曼方程的水动力极限的研究对于连接动力学理论和流体动力学这两个不同的学科是很重要的。一项严谨的研究将有助于理解大规模行为是如何从小尺度的动力学或结构中产生的。特别是对于弗拉索夫-麦克斯韦-玻尔兹曼系统，它将开辟一条新的研究路线。可压缩欧拉流体方程和纳维-斯托克斯流体方程以及玻尔兹曼动力学方程是气体动力学中最基本、最具挑战性的模型；它们在许多其他科学和工程领域也有广泛的应用。例如，欧拉方程和纳维-斯托克斯方程对研究浅层水波很有用，玻尔兹曼方程对半导体理论很重要。在这个项目中处理的数学问题不仅对这些方程的研究很重要，而且对物理学和其他科学也很重要。更好地理解它们将促进对更复杂、更现实的流的模型的研究。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The first goal of this project is to better understand physical vacuum as a free boundary for solutions to compressible Euler and Navier-Stokes equations with or without forcing and to develop a satisfactory nonlinear stability theory. The physical vacuum naturally appears in astrophysics and gas dynamics: a mathematical study of its behavior will help understand dynamics of gaseous stars. Furthermore, this investigation will lead to a general theory of degenerate hyperbolic equations. The second goal is to derive, with rigorous justification, various fluid equations from Boltzmann equations. Study of hydrodynamic limits for Boltzmann equations is important in connecting the two different subjects of kinetic theory and fluid dynamics. A rigorous study will help understand how large-scale behaviors emerge from dynamics or structures on small scales. In particular, for the Vlasov-Maxwell-Boltzmann system, it will open a new line of research. The compressible Euler and Navier-Stokes fluid equations and the Boltzmann kinetic equations are the basic, challenging models in gas dynamics; they have broad applications in many other areas of science and engineering as well. For example, Euler and Navier-Stokes equations are useful to study shallow water waves, and Boltzmann equations are important for semiconductor theory. The mathematical problems addressed in this project are not only important for the study of these equations, but also of interest in physics and other sciences. A better understanding of them would prompt the study of models for more complex, realistic flows.