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

物理和生物系统中重要平衡点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。玻尔兹曼方程是稀薄气体动力学理论的基础。众所周知，许多重要的流体方程都可以从玻尔兹曼方程形式上推导出来。我们建议用一个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、玻尔兹曼理论的Navier-Stokes近似在存在物理边界条件下的有效性以及二元流体模型中相分离的“前沿”解的稳定性。我们还建议研究各种物理和生物应用中的图案形成，例如在反应扩散系统和加热流体的Benard问题中。期望这些系统中的非线性不稳定模式可以用相应的线性系统在不稳定形成时间内的有限多个最快增长模式来表征。最后，我们建议进一步研究星系构型的非线性稳定性。动力学理论被用来描述大量稀薄粒子的动力学。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，也可以是巨大的物体，如星系中的恒星。这种稀薄的带电气体(等离子体)主导着我们的外层空间，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薄气体的长期动力学。此外，我们还建议研究星系模型的稳定性，并预测它们的长期动力学。图案形成在许多物理和生物系统中起着重要的作用。通过应用最近的不稳定性方法，我们建议发展一种数学理论来解释这些有趣的现象。

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.