Steady solutions of the Navier-Stokes equations in the plane
Steady solutions of the Navier-Stokes equations in the plane
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平面上纳维-斯托克斯方程的稳态解
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发表时间：
2015
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通讯作者：
J. Guillod
中科院分区：
文献类型：
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作者：
J. Guillod
本文研究了二维无界域上的不可压缩稳态Navier-Stokes方程。首先，对弱解的构造及其渐近行为的主要结果进行了回顾和结构化，以便所有的情况都可以用一种简洁的方式处理。大多数开放问题都与无限远处速度场消失的情况有关，这将是本研究剩余部分的主要主题。Navier-Stokes方程在零解周围的线性化导致Stokes方程在二维中是病态的。这就是著名的斯托克斯悖论，它指出，如果合力非零，斯托克斯方程的解将在无穷远处增长。通过研究Stokes方程与Navier-Stokes方程之间的联系，证明了即使净力消失，Navier-Stokes方程的速度场和压力场也不可能渐近于Stokes方程的速度场和压力场。然而，在某些情况下，速度场可以渐近于Stokes方程的两个精确解，这两个精确解也可以解Navier-Stokes方程。最后，基于物理论据，建立了具有非零净力的二维Navier-Stokes方程解在无穷远处的形式渐近展开式。该膨胀速度场的先导项以$|oldsymbol{x}|^{-1/3}$的形式衰减，并表现出尾迹行为。通过数值模拟验证了净力非零时的渐近展开，并分析了净力消失时的渐近行为。这表明，相对于Stokes线性化，Navier-Stokes方程允许其速度场在无穷远处趋于零的解，而且这表明可能的渐近线集合非常丰富。
This study is devoted to the incompressible and stationary Navier-Stokes equations in two-dimensional unbounded domains. First, the main results on the construction of the weak solutions and on their asymptotic behavior are reviewed and structured so that all the cases can be treated in one concise way. Most of the open problems are linked with the case of a vanishing velocity field at infinity and this will be the main subject of the remainder of this study. The linearization of the Navier-Stokes around the zero solution leads to the Stokes equations which are ill-posed in two dimensions. It is the well-known Stokes paradox which states that if the net force is nonzero, the solution of the Stokes equations will grow at infinity. By studying the link between the Stokes and Navier-Stokes equations, it is proven that even if the net force vanishes, the velocity and pressure fields of the Navier-Stokes equations cannot be asymptotic to those of the Stokes equations. However, the velocity field can be in some cases asymptotic to two exact solutions of the Stokes equations which also solve the Navier-Stokes equations. Finally, a formal asymptotic expansion at infinity for the solutions of the two-dimensional Navier-Stokes equations having a nonzero net force is established based physical arguments. The leading term of the velocity field in this expansion decays like $|oldsymbol{x}|^{-1/3}$ and exhibits a wake behavior. Numerical simulations are performed to validate this asymptotic expansion when is net force is nonzero and to analyze the asymptotic behavior in the case where the net force is vanishing. This indicates that the Navier-Stokes equations admit solutions whose velocity field goes to zero at infinity in contrast to the Stokes linearization and moreover this shows that the set of possible asymptotes is very rich.