Asymptotic Problems in Fluid Mechanics and Gas Dynamics

这个建议是关于流体力学和气体动力学中的一些渐近问题的研究。当描述某种类型的流体(或任何其他物理系统)运动的方程中的无量纲参数epsilon趋于零时，就会出现渐近问题。当我们试图证明达到极限时，会遇到许多数学问题，这些问题主要是由于方程类型的变化，许多空间和时间尺度的存在，边界层的存在(我们不能再对初始系统和极限系统施加相同的边界条件)，以及高频时间振荡的存在。在这个建议中，PI打算研究Boltzmann方程的流体力学极限，即可压缩-不可压缩极限，这些渐近问题允许我们在极限处得到更简单的模型，因为我们通常有更少的变量或(和)更少的未知数。这简化了数值模拟，提高了我们对参数较小时的普遍现象的理解，实际上，我们可以不求解初始系统，而是求解极限系统，然后添加一个校正器。

This proposal is about the study of some asymptotic problemsin Fluid Mechanics and Gas Dynamics. Asymptotic problems arise when a dimensionless parameter epsilon goes to zero in an equation describing the motion of some type of fluid (or any other physical system). Many mathematical problems are encountered when we try to justify the passage to the limit, which are mainly due to the change of the type of the equations, the presence of many spatial and temporal scales, the presence of boundary layers (we can no longer impose the same boundary conditions for the initial system and the limit one), the presence of oscillations intime at high frequency .... In this Proposal, the PI intends to study (among other problems)the hydrodynamic limit of the Boltzmann equation, the compressible-incompressible limit, These asymptotic problems allow us to get simpler models at the limit, due to the fact that we usually have fewer variables or (and) fewer unknowns. This simplifies the numerical simulations and improves our understanding of the prevailing phenomenon when the parameter is small, in fact, instead of solving the initial system, we can solve the limit system and then add a corrector.