Title: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory Show Chinese title

Title: 希尔伯特的第六问题：通过玻尔兹曼的气体动理论推导流体方程

Abstract: In this paper, we rigorously derive the fundamental PDEs of fluid mechanics, such as the compressible Euler and incompressible Navier-Stokes-Fourier equations, starting from the hard sphere particle systems undergoing elastic collisions. This resolves Hilbert's sixth problem, as it pertains to the program of deriving the fluid equations from Newton's laws by way of Boltzmann's kinetic theory. The proof relies on the derivation of Boltzmann's equation on 2D and 3D tori, which is an extension of our previous work (arXiv:2408.07818). Show Chinese abstract

Abstract: 在本文中，我们从经历弹性碰撞的硬球粒子系统出发，严格推导出流体力学的基本偏微分方程，如可压缩欧拉方程和不可压缩纳维-斯托克斯-傅里叶方程。 这解决了与通过玻尔兹曼统计理论从牛顿定律推导流体方程程序相关的希尔伯特第六问题。 该证明依赖于在二维和三维环面上推导玻尔兹曼方程，这是我们在之前工作（arXiv:2408.07818）的扩展。