We consider the 2D hyperviscosity equations on $\mathbb{T}\times \mathbb{R}$. We show that if the initial data of 2D hyperviscosity equations are ϵ‐close to the shear flows (U(y),0), which are sufficiently small perturbations of Couette flow (y,0), then the solution will stay ϵ‐close to $(e^{-\nu t{\partial }_y^4}U(y),0)$ for all t>0, where $ϵ\ll {\nu }^{\frac{1}{2}}$ and ν denotes the kinematic viscosity coefficient. What is more, by the mixing‐enhanced effect, the solutions converge to decaying shear flows for $t\gg {\nu }^{-\frac{1}{5}}$, which is faster than the heat‐equation timescale. Hence, we conclude that the stability threshold of 2D hyperviscosity equations with initial data (U(y),0) is not worse than ${\nu }^{\frac{1}{2}}$.

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二维接近库埃特流的剪切流的高粘度方程的Sobolev稳定性阈值

我们考虑了二维超粘方程 $\mathbb{Ť}\times \mathbb{[R}$。我们表明，如果2D高粘方程的初始数据是ε -close到剪切流（û（Ý），0），其是充分Couette流（的小扰动Ý，0），然后将溶液将保持ε -close至$（{Ë}^{-\nu Ť{\partial }_{ÿ}^4}ü（ÿ），0）$对于所有t > 0，$ϵ\ll {\nu }^{\frac{\mathrm{1个}}{2}}$并且ν表示的运动粘度系数。而且，通过混合增强效应，解收敛到了$Ť\gg {\nu }^{-\frac{\mathrm{1个}}{5}}$，这比热平衡时间表要快。因此，我们得出结论：具有初始数据（U（y），0）的2D高粘度方程的稳定性阈值不小于${\nu }^{\frac{\mathrm{1个}}{2}}$。