Asymptotic Problems in Fluid Mechanics, Gas Dynamics and Quantum Mechanics

题目：流体力学、气体动力学和量子力学中的渐近问题。摘要首席研究员（PI）建议继续研究来自流体力学、气体动力学和量子力学的一些渐近问题。当无量纲参数（如雷诺数、马赫数或时间标度）趋近于零或无穷大时，就会出现这些渐近问题。在研究这些问题时，出现了许多数学上的困难。这些困难主要是由于方程类型的变化、许多时空尺度的存在、共振的存在、边界层的存在……为了克服这些困难，已经开发了许多工具，例如引入不同类型的度量来描述强收敛性的缺陷，使用补偿紧性类型的参数，使用平均引理，使用能量方法和相对熵方法……当然，未来还应该开发许多其他工具。更具体地说，PI计划继续研究玻尔兹曼方程的流体动力极限，当Knudsen数趋于0时，特别是在有界域中，并将其与可压缩-不可压缩极限中的一些已知结果联系起来。他还计划在水波问题中研究韦伯数趋于无穷时的极限。PI提出研究流体力学、气体力学和量子力学中的一些渐近问题。这些渐近问题的研究对于更好地理解无极限情况下复杂系统的行为具有重要意义。这也使我们能够更好地理解正在发生的真实物理现象。此外，它还提供了一种对不同物理模型进行严格推导的方法。它还可以更好地了解每个简化模型的有效域。对于工程师和物理学家来说，这是非常重要的，他们正在寻找最简单的模型来捕捉现象，以实现数字或应用于现实生活。

Proposal DMS-0403983PI: Nader Masmoudi, Courant Institute-NYUTitle: Asymptotic problems in fluid mechanics, gas dynamics and quantum mechanicsABSTRACTThe Principal Investigator (PI) proposes to continue working onsome asymptotic problems coming from Fluid Mechanics, GasDynamics and Quantum Mechanics. These asymptotic problems arisewhen a dimensionless parameter (such as the Reynolds number, theMach number or a time scaling) goes to zero or to infinity. Instudying these problems, many mathematical difficulties arise.These difficulties are mainly due to the change of the type ofthe equations, the presence of many temporal and spatial scales,the presence of resonances, the presence of boundary layers ...Many tools have been developed to circumvent these difficultiessuch as the introduction of different types of measures todescribe the defect of strong convergence, the use of compensated compactness type arguments, the use of averaging lemma, the use ofenergy methods and relative entropy methods ... Of course, manyother tools should be developed in the future. More specificallythe PI plans to continue working on the hydrodynamic limits ofthe Boltzmann equation when the Knudsen number goes to 0especially in bounded domains and relate this to some of the knownresults in the compressible-incompressible limit. He also plans tolook at the limit when the Weber number goes to infinity in thewater wave problem.The PI proposes to study some asymptotic problems coming fromFluid Mechanics, Gas Dynamics and Quantum Mechanics. The study ofthese asymptotic problems is very important to get a betterunderstanding about the behavior of complicated systems indifferent limiting cases. This also allows to have a betterunderstanding of the real physical phenomenon taking place.Moreover, it provides a way of giving rigorous derivations ofdifferent physical models. It also gives a better knowledge aboutthe domain of validity of each simplifying model. This is veryimportant for engineers and physicists who are looking for thesimplest model that captures the phenomenon to implementnumerically or to apply in real life.