In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary. The novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticity that are uniform in the inviscid limit.

中文翻译：

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剪切剖面周围半空间中线性化纳维斯托克斯方程的求解的锐界

在本文中，我们在半平面和半空间（$\mathbb{R}_+^2$ 或 $\mathbb{ R}_+^3$)，使用狄利克雷边界条件，假设该剪切层对于欧拉方程在光谱上不稳定。在无粘性极限中，由于规定的无滑移边界条件，涡度在边界附近变得无界。本文的新颖之处在于引入了捕捉无界涡度的边界层范数，并在无粘性极限中对这种涡度进行了精确估计。