Asymptotic Problems with Boundary Effect in Kinetic Theory

动力学理论中边界效应的渐近问题

• 批准号: 1853002
• 负责人: Lei Wu
• 金额: $ 9.14万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-15 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1853002&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Boundary; Effect; Kinetic

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.

动力学理论描述了大量颗粒的动力学，例如经过翼型的空气颗粒流或中子在核反应堆中的碰撞。动力学描述以统计方式桥接了牛顿力学对粒子运动的微尺度建模，以及通过连续流体力学的宏观尺度建模。该研究项目旨在开发新颖的数学方法，以定量地表征这些多尺度模型。该研究的重点是在周围环境的影响下在空间界面区域内的稀有气体，中子，电子和离子的运动。 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.在有限的域中，动力学边界校正（即边界层）起着至关重要的作用。研究人员打算在存在奇异边界层的情况下证明渐近近似的有效性是合理的。此外，考虑了初始的相互作用和非线性效应。为了应对非标准的渐近扩展，研究者试图在边界层中开发一般的几何校正理论，剩余的估计值，米尔恩规律性分析，非本地能量方法和边界层分解。这些技术可以进一步应用于Vlasov系统和磁流失动力学（MHD）方程。同样，涉及分数扩散和随机扩散的严格衍生。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。

项目成果

Asymptotic analysis of unsteady neutron transport equation

非定常中子输运方程的渐近分析

• DOI: 10.1002/mma.5531
• 发表时间: 2019
• 期刊: Mathematical Methods in the Applied Sciences
• 影响因子: 2.9
• 作者: Wu, Lei

Dynamics and stability of sessile drops with contact points

带接触点的固着液滴的动力学和稳定性

• DOI: 10.1016/j.jde.2020.10.012
• 发表时间: 2021
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Tice, Ian;Wu, Lei
• 通讯作者: Wu, Lei

Diffusive Limit of Transport Equation in 3D Convex Domains

3D凸域中输运方程的扩散极限

• DOI: 10.1007/s42543-020-00032-4
• 发表时间: 2021
• 期刊: Peking Mathematical Journal
• 作者: Wu, Lei

Dynamics and Stability of Surface Waves with Bulk-Soluble Surfactants

体溶性表面活性剂表面波的动力学和稳定性

• DOI: 10.1007/s10440-018-0203-0
• 发表时间: 2019
• 期刊: Acta Applicandae Mathematicae
• 影响因子: 1.6
• 作者: Tice, Ian;Wu, Lei
• 通讯作者: Wu, Lei

Hydrodynamic Limit of 3dimensional Evolutionary Boltzmann Equation in Convex Domains

凸域中三维演化玻尔兹曼方程的流体力学极限

• DOI: 10.1137/20m1375735
• 发表时间: 2022
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Wu, Lei;Ouyang, Zhimeng
• 通讯作者: Ouyang, Zhimeng