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

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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.