The goal of this paper is to present the recent development of mathematical fluid dynamics in the framework of classical continuum mechanics phenomenological models. In particular, we discuss the Navier–Stokes (viscous) and the Euler (inviscid) systems modeling the motion of a compressible fluid. The theory is developed from fundamental physical principles, the necessary mathematical tools introduced at the moment when needed. In particular, we discuss various concepts of solutions and their relevance in applications. Particular interest is devoted to well-posedness of the initial-value problems and their approximations including possibly certain numerical schemes.

中文翻译：

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运动流体的数学理论

本文的目的是在经典连续力学力学现象学模型的框架内，介绍数学流体动力学的最新发展。特别是，我们讨论了对可压缩流体运动进行建模的Navier-Stokes（粘液）和Euler（无粘液）系统。该理论是从基本物理原理发展而来的，必要时会引入必要的数学工具。特别是，我们讨论了解决方案的各种概念及其在应用程序中的相关性。特别关注的是初值问题的适定性及其近似值，包括可能的某些数值方案。