On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows
On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows
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DOI：
10.1016/j.matpur.2024.02.001
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发表时间：
2017-06
期刊：
影响因子：
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通讯作者：
E. Grenier;Toan T. Nguyen
中科院分区：
文献类型：
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作者：
E. Grenier;Toan T. Nguyen
在1904年，普朗特引入了他著名的边界层，以描述当粘性变为0时，Navier Stokes方程在边界附近的解的行为。他的解释后来被RE Caflisch和M.萨马蒂诺本文证明了在Sobolev正则解的情形下，甚至在Sobolev空间中Prandlt方程适定的情形下，他的展开式在L∞范数下直到O（ν 1/4）阶项都是假的.此外，我们还证明了当ν= 0时稳定的单调边界层廓线在ν> 0时是非线性不稳定的，只要ν足够小，在L∞范数下达到O（ν 1/4）项。
In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by RE Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O (ν 1/4) order terms in L∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when ν= 0, are nonlinearly unstable when ν> 0, provided ν is small enough, up to O (ν 1/4) terms in L∞ norm.