论文标题
一种动态的方法，可以研究近距离流动的不稳定性
A dynamical approach to the study of instability near Couette flow
论文作者
论文摘要
在本文中，当扰动位于关键空间$ h^1_xl_y^2 $时，我们获得了带有小粘度的Navier-Stokes方程的Couette流的最佳不稳定性阈值。 More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $ν^{\frac{1}{2}-δ_0}$ with any small $δ_0>0$, which implies that $ν^{\frac{1}{2}}$ is the sharp stability threshold.在我们的方法中，我们证明了瞬态指数生长，而无需提及特征值或伪谱。作为应用程序，对于附近近距离流的剪切流的线性化欧拉方程，我们提供了一种新工具来证明存在相应的雷利操作员的增长模式，并提供特征值的精确位置。
In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity $ν>0$, when the perturbations are in the critical spaces $H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $ν^{\frac{1}{2}-δ_0}$ with any small $δ_0>0$, which implies that $ν^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.
