Enhanced Dissipation in the Navier–Stokes Equations Near the Poiseuille Flow
Enhanced Dissipation in the Navier–Stokes Equations Near the Poiseuille Flow
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DOI：
10.1007/s00220-020-03814-0
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发表时间：
2019-01
影响因子：
2.4
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
我们考虑二维Navier-Stokes方程在小粘性的Poiseuille流附近的解。我们的第一个结果涉及线性化问题的一个半群估计。这里我们证明了线性解的依赖于x的模式在时间尺度上衰减，与之成正比。这种效应通常被称为增强的耗散或亚稳定，因为它比常规的耗散时间标度(这也是与x无关的模式自然衰变的时间标度)衰减得快得多。我们使用一种改编的亚矫顽力方法来实现这一点。我们的第二个结果涉及完全的非线性方程。我们证明了，当Poiseuille流的扰动最初至多是大小时，它将一直如此。此外，增强的耗散也在这种情况下持续存在，因此解的x相关模式在一个数量级的时间尺度上耗散。为了处理区域和泊松流动本身的无界性，通过使用半群估计和对非线性项的仔细分析，通过Bootstrap引理建立了这个转变阈值。
We consider solutions to the 2d Navier–Stokes equations onclose to the Poiseuille flow, with small viscosity. Our first result concerns a semigroup estimate for the linearized problem. Here we show that thex-dependent modes of linear solutions decay on a time-scale proportional to. This effect is often referred to asenhanced dissipationormetastabilitysince it gives a much faster decay than the regular dissipative time-scale(this is also the time-scale on which thex-independent mode naturally decays). We achieve this using an adaptation of the method of hypocoercivity. Our second result concerns the full nonlinear equations. We show that when the perturbation from the Poiseuille flow is initially of size at most, then it remains so for all time. Moreover, the enhanced dissipation also persists in this scenario, so that thex-dependent modes of the solution are dissipated on a time scale of order. This transition threshold is established by a bootstrap argument using the semigroup estimate and a careful analysis of the nonlinear term in order to deal with the unboundedness of the domain and the Poiseuille flow itself.