论文标题
Kolmogorov和Poiseuille在2D Euler方程中流动附近的固定结构
Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations
论文作者
论文摘要
我们研究了在两个具有关键点的规范剪切流附近的不可压缩$ 2D $ EULER方程的解决方案的行为，即Kolmogorov和Poiseuille流动，对相关的Navier-Stokes问题产生了影响。 我们展示了一个新的，新的，非平凡的固定状态的分析规律性，它们随时接近Square torus $ \ mathbb {t}^2 $上的kolmogorov流。这种情况与某些单调剪切流的设置形成鲜明对比，例如COUETTE流动：在这两种情况下，线性化问题都表现出一种“无粘性阻尼”机制，从而导致基本流动的扰动放松回到附近的剪切流。尽管这种效果持续到非线性，对于某些单调剪切流的适当小且规则的扰动，但对于Kolmogorov流动，我们的结果表明这是不可能的。 我们对这些固定状态的构建建立在$ \ mathbb {t}^2 $上的kolmogorov流的全球结构中的变性。在这方面，矩形圆环上的kolmogorov和通道中的poiseuille流都大不相同，我们表明，即使在相对较低的规律性$ h^3 $ resp中，附近唯一的固定状态确实必须是剪切。 $ h^{5+} $。 此外，我们表明这种行为在相关的Navier-Stokes设置中紧密反映：Poiseuille和Kolmogorov附近的线性化问题都表现出增强的耗散速度。我们和其他人的先前工作表明，这种效果在Poiseuille流的完整的非线性问题中幸存下来，并且只要扰动位于一定阈值以下，则在矩形Tori上的Kolmogorov流附近。但是，我们在这里表明，相应的结果无法在$ \ mathbb {t}^2 $上的Kolmogorov流附近保持。
We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $\mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an "inviscid damping" mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on $\mathbb{T}^2$. In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity $H^3$ resp. $H^{5+}$. In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on $\mathbb{T}^2$.
