Asymptotic problems and Well-posedness results in Fluid Mechanics and Plasma Physics

本项目将继续研究流体力学、气体动力学和等离子体物理中的一些渐近问题，以及研究自由边界问题和非牛顿流体中的适定性问题。当一个无量纲参数(如雷诺数或电子热速度与光速之比)变为零或无穷大时，就会出现这些渐近问题。在研究这些问题时，出现了许多数学困难。这些困难主要是由于方程类型的改变、许多时间和空间尺度的存在、共振的存在、边界层的存在等。为了规避这些困难，已经开发了许多工具，例如引入不同类型的措施来描述强收敛的缺陷，使用补偿紧性类型的论点，使用平均引理，使用能量方法和相对熵方法，等等。在这项工作中将开发新的工具。这些问题的研究对于更好地理解复杂系统在不同极限情况下的行为是非常重要的。这也使我们能够更好地理解正在发生的真实物理现象。它还提供了关于每个简化模型的有效性领域的更好的知识。这对于工程师和物理学家来说是非常重要的，他们正在寻找最简单的模型来捕捉现象，以便在数值上实现或应用于现实生活。其中一个问题是当Knudsen数为0时Boltzmann方程的流体动力学极限，特别是在有界域中。形式上，极限方程(欧拉系统，纳维斯托克斯，...)取决于其他参数，例如马赫数、雷诺数和时间比例。特别是，该项目的目标之一是了解流体边界条件与动力学边界条件。这是理解流体方程和动力学方程边界上的耦合的一步。当然，当我们必须从使用动力学模型的区域转向使用流体模型的区域时，这对于数值模拟来说是一个非常重要的问题。此外，这个项目的另一个目标是研究一些非牛顿流体，特别是聚合物液体(例如蛋清、血液或面团)。这些体系需要流体和聚合物之间的耦合，在应用物理、化学和生物学的许多分支中都引起了极大的兴趣。

This project is to continue working on some asymptotic problems in Fluid Mechanics, Gas Dynamics and Plasma physics, as well as to study some well-posedness questions arising in free boundary problems and non-Newtonian fluids. These asymptotic problems arise when a dimensionless parameter (such as the Reynolds number or the ratio between the electron thermal speed and the speed of light) goes to zero or to infinity. In studying these problems, many mathematical difficulties arise. These difficulties are mainly due to the change of the type of the 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 difficulties such as the introduction of different types of measures to describe the defect of strong convergence, the use of compensated compactness type arguments, the use of averaging lemma, the use of energy methods and relative entropy methods, ... . New tools will be developed in this work. Hopefully, they can be applied to other areas of mathematics.The study of these problems is very important to get a better understanding about the behavior of complicated systems in different limiting cases. This also allows an improved understanding of the real physical phenomena taking place. It also gives a better knowledge about the domain of validity of each simplifying model. This is very important for engineers and physicists who are looking for the simplest model that captures the phenomena to implement numerically or to apply in real life. One of these problems is the hydrodynamic limit of the Boltzmann equation when the Knudsen number goes to 0 especially in bounded domains. Formally, the limit equation (Euler system, Navier-Stokes, ...) depends on additional parameters such as the Mach number, the Reynolds number and the time scaling. In particular, one of the goals of the project is to understand fluid boundary conditions from kinetic ones. This is a step towards understanding the coupling at the boundary between fluid equations and kinetic equations. Of course this is a very important question for numerical simulations when we have to go from a domain where a kinetic model is being used to a domain where a fluid model is being used. Besides, another goal of this project is the study of some non-Newtonian fluids and especially polymeric liquids (egg white, blood or dough for example). These systems require a coupling between fluids and polymers and are of great interest in many branches of applied physics, chemistry and biology.