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

动力学理论稳定性研究的偏微分方程方法及其应用

• 批准号: 0603815
• 负责人: Yan Guo
• 金额: $ 24.21万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603815&HistoricalAwards=false
• 关键词: PDE; Methods; Stability; Study; Kinetic

The study of nonlinear stability and instability of important equilibria in physical and biological systems ultimately relies on rigorous analytical proofs. The Boltzmann equation is the foundation in the kinetic theory for dilute gases. It is well known that many important fluid equations can be formally derived from the Boltzmann equation. We propose to use a nonlinear energy method to prove the validity of diffusive expansion in linear neutron transport theory, of the Navier-Stokes approximation of the Boltzmann theory in the presence of physical boundary conditions, and of the stability of `front' solution for phase segregation in a binary fluid model. We also propose to study pattern formation in various physical and biological applications such as in reaction-diffusion systems and the Benard problem for a heated fluid. It is expected that the pattern of nonlinear instabilities in these system can be characterized by the finitely many fastest growing modes for the corresponding linear system, over the time of instability formation. Finally, we propose to further study nonlinear stability of galaxy configurations.Kinetic theory is used to describe the dynamics of a large number of dilute `particles'. These `particles' can be as small as gas molecules or charged ions or electrons in a plasma, or enormous objects such as stars in galaxies. Such kind of dilute charged gases (plasma) dominates our outer space, and plays the crucial role in our fusion research. We propose to study the long-time dynamics of these dilute gases form a mathematical standpoint. Furthermore, we propose to study stability of the galaxy models and predict their long-time dynamics. Pattern formation plays an important role in many physical and biological systems. By applying a recent instability method, we propose to develop a mathematical theory to explain these interesting phenomena.

对物理和生物系统中重要平衡的非线性稳定性和不稳定性的研究最终依赖于严格的分析证明。玻尔兹曼方程是稀气体动力学理论的基础。众所周知，许多重要的流体方程可以从玻尔兹曼方程形式上推导出来。我们提出用一种非线性能量方法来证明线性中子输运理论中扩散展开的有效性，玻尔兹曼理论中存在物理边界条件的Navier-Stokes近似的有效性，以及二元流体模型中相偏析“前”解的稳定性。我们还建议研究各种物理和生物应用中的模式形成，例如反应扩散系统和加热流体的贝纳德问题。预计这些系统的非线性不稳定性模式可以用相应线性系统在不稳定性形成时间内的有限多个最快增长模态来表征。最后，我们提出进一步研究星系结构的非线性稳定性。动态论是用来描述大量稀“粒子”的动力学。这些“粒子”可以小到气体分子或等离子体中的带电离子或电子，也可以是像星系中的恒星这样的巨大物体。这种稀薄的带电气体（等离子体）支配着我们的外层空间，在我们的聚变研究中起着至关重要的作用。我们建议从数学的角度来研究这些稀释气体的长期动力学。此外，我们建议研究星系模型的稳定性并预测它们的长期动力学。模式形成在许多物理和生物系统中起着重要作用。通过应用最近的不稳定性方法，我们提出了一个数学理论来解释这些有趣的现象。