Applied Analysis on the Navier-Stokes Equations and Related Dynamical Systems

Navier-Stokes方程和Euler方程是流体力学的主要方程。我们寻找奇点和伪奇点。Proudman。研究了带有积分约束的Johnson方程和反应扩散方程。通过数值计算，我们发现存在一个拓扑简单的解，它们只存在于二维Navier-Stokes方程和与之密切相关的方程中。计算了流体粒子在连续水波中的运动轨迹。所谓的斯托克斯漂移是用数学方法证明的。我们的证明比其他任何证明都简单，而且易于适用于数值计算。我们的数值方法不仅可以计算重力波，而且可以计算毛细管重力波。

The Navier-Stokes equations and the Euler equations are the master equations of the fluid mechanics. We searched for singularity and pseudo singularity. The Proudman. Johnson equation and reaction-diffusion equation with integral constraint were studied. By numerically computing solutions, we found that a topologically simple solutions exists and they exist only in two-dimensional Navier-Stokes equations and those closely related to it.Trajectories of fluid particles in a progressive water-waves were computed. What is called the Stokes drift was proved mathematically. Our proof is simpler than any other proof and easily adapted to the numerical computation. Our numerically method enables us to compute not only the gravity waves but also capillary-gravity waves.