PDE Methods in Kinetic Theory and Their Applications

动力学理论是多尺度模拟的核心，它将微观颗粒模型与宏观流体模型联系起来。在运动学理论中有许多具有挑战性的开放问题，这些问题无论从数学还是物理角度来说都是非常重要的。这项研究的主要目标是继续发展新的数学方法来解决偏微分方程组中的公开问题，这些问题出现在运动论和数学物理的其他领域。这些研究将包括：稀薄气体玻尔兹曼理论中的边界效应，从动力学理论推导各种宏观流体模型，以及在广泛的应用问题中稳态的非线性稳定性和不稳定性。这些研究项目将在物理科学的许多领域产生重要影响。Vlasov理论(无碰撞玻尔兹曼理论)中稳定平衡的研究将为核聚变中的等离子体控制和星系演化提供新的线索；Stefan问题的研究将为晶体生长的形态稳定性和材料科学中出现的许多其他问题建立数学基础；Vlasov-Boltzmann模型中的相变研究将有助于更好地理解二元流体中的相分离。该项目的一个重要目标是为参与开展这项研究的学生和初级科学家提供培训。

Kinetic theory is at the center of multi-scale modeling, which connects the microscopic particle models to macroscopic fluid models. There are many challenging open problems in kinetic theory which are of great importance from both mathematical and physical standpoints. The main goal of this research is to continue developing new mathematical methods to resolve open problems in partial differential equations arising in the kinetic theory and other fields in mathematical physics. The investigations will include: boundary effects in the Boltzmann theory for dilute gases, derivation of various macroscopic fluid models from the kinetic theory, and nonlinear stability and instability of steady states in a wide range of applied problems. These research projects will have important impacts in many areas of physical sciences. The study of stable equilibria in the Vlasov theory (collisionless Boltzmann theory) will shed new light on plasma control in nuclear fusion and on galaxy evolution; the study of the Stefan problem will build a mathematical foundation for morphological stability of crystal growth and many other problems arising in materials sciences; and the study of phase-transitions in the Vlasov-Boltzmann model will lead to better understanding of phase segregation in binary fluids. An important objective of the project is to provide training for students and junior scientists involved in carrying out this research.