Asymptotic Problems with Boundary Effect in Kinetic Theory

动力学理论描述了大量粒子的动力学，如空气粒子通过翼型的流动，或核反应堆中中子的碰撞。动力学描述以一种统计的方式，将牛顿力学对颗粒运动的微观建模与连续介质流体力学的宏观建模联系起来。本研究项目旨在开发新的数学方法来定量描述这些多尺度模型。研究的重点是在周围环境的影响下，稀薄气体、中子、电子和离子在空间有界区域内的运动。它的应用范围从半导体或核聚变等高科技领域，到日常生活中的喷水器或荧光灯等设备。具体地说，该项目专注于Boltzmann方程或输运方程的流体动力学极限，即当一个小参数，如克努森数或斯特鲁哈尔数接近零时，解如何渐近变化。在有界域中，动力学边界修正(即边界层)起着至关重要的作用。研究者打算证明在存在奇异边界层的情况下渐近近似的有效性。此外，还考虑了初始-边界相互作用和非线性效应。为了处理非标准的渐近展开式，研究者试图发展边界层几何校正、重标度余量估计、Milne正则性分析、非局部能量方法和边界层分解的一般理论。这些技术可以进一步应用于Vlasov系统和磁流体动力学(MHD)方程。此外，还涉及分数扩散和随机扩散的严格推导。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Kinetic theory describes the dynamics of a large number of particles, such as flows of air particles passing an airfoil, or neutrons' collisions in a nuclear reactor. In a statistical manner, the kinetic description bridges the micro-scale modeling of motion of particles by Newtonian mechanics and the macro-scale modeling by continuum fluid mechanics. This research project aims to develop novel mathematical methods to quantitatively characterize these multi-scale models. The investigation focuses on the motion of rarefied gas, neutrons, electrons, and ions, in spatially bounded regions under the influence of the surrounding environment. Its applications range from high-tech fields like semi-conductors or nuclear fusion, to daily-life devices like water sprays or fluorescent lamps.Specifically, this project concentrates on the hydrodynamic limit of the Boltzmann equation or transport equation, i.e. how the solution varies asymptotically when a small parameter, either the Knudsen number or the Strouhal number, approaches zero. In bounded domains, kinetic boundary corrections (i.e. boundary layers), play a crucial role. The investigator intends to justify the validity of the asymptotic approximation in the presence of singular boundary layers. Moreover, the initial-boundary interactions and nonlinear effects are taken into consideration. To tackle the non-standard asymptotic expansions, the investigator seeks to develop general theories of geometric correction in boundary layers, rescaled remainder estimates, Milne regularity analysis, non-local energy methods, and boundary layers decomposition. These techniques can be further applied to Vlasov systems and magnetohydrodynamical (MHD) equations. Also, the rigorous derivations of fractional diffusion and stochastic diffusion are involved.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.