论文标题
单调粘性边界层中非线性不稳定性的发作
Onset of nonlinear instabilities in monotonic viscous boundary layers
论文作者
论文摘要
在本文中，我们研究了边界附近的Navier Stokes方程的剪切层曲线的非线性稳定性。在粘度为$ 0 $的情况下，这个问题在研究有限域中Navier Stokes方程的无粘性极限的研究中起着重要作用。 Navier Stokes方程的剪切层的稳定性取决于其对Euler方程的稳定性。如果Euler线性不稳定，那么众所周知，对于Navier Stokes方程式而言，它也是不稳定的，前提是粘度足够小：初始扰动会增长，直到达到$ o（1）$ in $ l^\ l^\ inftty $ norm。如果对Euler的线性稳定，则情况更为复杂，因为粘性不稳定要慢得多，并且仅在第一种情况下而不是$ o（1）$的订单$ O（ν^{ - 1/4}）$（而不是$ o（1）$）。目前尚不清楚线性不稳定性是否完全发展，直到它们达到阶数$ o（1）$，还是被非线性降低并以较小的幅度饱和，或者订购$ O（ν^{1/4}）$（例如）。 在本文中，我们研究了立方相互作用对线性不稳定性生长的影响。在指数曲线和Blasius轮廓的情况下，我们获得了非线性驯服线性不稳定性。因此，我们猜想小扰动会增长，直到它们达到幅度$ o（ν^{1/4}）$，从而在边界附近的临界层中形成小卷。这个猜想的数学证明是开放的。
In this paper we study the nonlinear stability of a shear layer profile for Navier Stokes equations near a boundary. This question plays a major role in the study of the inviscid limit of Navier Stokes equations in a bounded domain as the viscosity goes to $0$. The stability of a shear layer for Navier Stokes equations depends on its stability for Euler equations. If it is linearly unstable for Euler, then it is known that it is also nonlinearly unstable for Navier Stokes equations provided the viscosity is small enough: an initial perturbation grows until it reaches $O(1)$ in $L^\infty$ norm. If it is linearly stable for Euler, the situation is more complex, since the viscous instability is much slower, with growth rates of order $O(ν^{-1/4})$ only (instead of $O(1)$ in the first case). It is not clear whether linear instabilities fully develop till they reach a magnitude of order $O(1)$ or whether they are damped by the nonlinearity and saturate at a much smaller magnitude, or order $O(ν^{1/4})$ for instance. In this paper we study the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile and Blasius profile we obtain that the nonlinearity tame the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude $O(ν^{1/4})$ only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open.
