Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations
Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations
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DOI：
10.1007/s00205-023-01842-3
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发表时间：
2020-07
影响因子：
2.5
通讯作者：
Michele Coti Zelati;T. Elgindi;Klaus Widmayer
中科院分区：
文献类型：
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作者：
Michele Coti Zelati;T. Elgindi;Klaus Widmayer
我们研究了不可压缩二维欧拉方程的解的行为附近的两个典型的剪切流的临界点，Kolmogorov和Poiquilille流，与相关的Navier-Stokes问题的后果。我们展示了一个大家庭的新的，非平凡的定态，任意接近Kolmogorov流的平方torusin解析规律。这种情况与某些单调剪切流的设置形成强烈对比，例如库埃特流：线性化问题表现出一种“无粘阻尼”机制，导致基流的扰动松弛回到附近的剪切流。我们的结果表明，这样一个简单的描述的长时间的行为是不可能的解决方案附近的Kolmogorov流on. Our建设的新的定态建立在一个退化的全局结构的Kolmogorov流上，我们也表现出缺乏对应的线性化描述的定态集和其真正的非线性结构。矩形环面上的Kolmogorov流和槽道中的Poiquilille流是非常不同的。我们表明，唯一的定态附近，他们必须确实是剪切，即使在相对较低的规则性。此外，我们表明，这种行为是密切反映在相关的Navier-Stokes设置：附近的Poietille和Kolmogorov流的线性化问题都表现出增强的耗散率。我们和其他人以前的工作表明，这种效应在Poietille流附近和矩形环面上的Kolmogorov流附近的完整的非线性问题中仍然存在，只要扰动低于一定的阈值。然而，我们在这里表明，相应的结果不能保持附近的Kolmogorov流。
We study the behavior of solutions to the incompressible 2dEuler 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 that are arbitrarily close to the Kolmogorov flow on the square torusin analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the long-time behavior is not possible for solutions near the Kolmogorov flow on. Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on, and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. 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.