标题： 二维Navier边界条件下接近Couette剪切流的稳定性阈值 显示英文标题

标题： Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D

摘要： 在这项工作中，我们证明了定义在周期通道上的二维Navier-Stokes方程的阈值定理，即$\mathbb{T} \times [-1,1]$，附加Navier边界条件$\omega|_{y = \pm 1} = 0$。初始数据被视为Couette流的扰动，具体来说：假设扰动的剪切分量足够小（在适当的Sobolev空间中），但重要的是它与$\nu$无关。另一方面，非零模式假设在各向异性Sobolev空间中具有大小$O(\nu^{\frac12})$。对于此类数据，我们证明了由此产生的解具有非线性增强耗散和无粘性阻尼。主要创新在于定量捕捉\textit{无粘性阻尼}，为此我们引入了一个新的奇异积分算子（SIO），它是通常用于证明阻尼的Fourier乘子在物理空间中的类比。然后我们将此SIO纳入非线性次耗散框架中。 显示英文摘要

摘要： In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $\mathbb{T} \times [-1,1]$, supplemented with Navier boundary conditions $\omega|_{y = \pm 1} = 0$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $\nu$. On the other hand, the nonzero modes are assumed size $O(\nu^{\frac12})$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework.