First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar states, have been numerically observed as one type of metastable states in the study of 2D turbulence. For both rectangular and square tori, we prove that the non-shear part of perturbations near Kolmogorov flow decays in a time scale much shorter than the viscous time scale. The results are obtained for both the linearized NS equations with any initial vorticity in L 2 , and the nonlinear NS equation with initial L 2 norm of vorticity of the size of viscosity. In the proof, we use the Hamiltonian structure of the linearized Euler equation and RAGE theorem to control the low frequency part of the perturbation. Second, we consider two classes of shear flows for which a sharp stability criterion is known. We show the inviscid damping in a time average sense for non-shear perturbations with initial vorticity in L 2 . For the unstable case, the inviscid damping is proved on the center space. Our proof again uses the Hamiltonian structure of the linearized Euler equation and an instability index theory recently developed by Lin and Zeng for Hamiltonian PDEs.

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Nanopore technology is a burgeoning detection technology for single-molecular sensing and ion rectification. Solid-state nanopores have attracted more and more attention because of their higher stability and tunability than biological nanopores. However, solid-state nanopores still suffer the drawbacks of low signal-to-noise ratio and low resolution, which hinder their practical applications. Thus, developing operatical and useful methods to overcome the shortages of solid-state nanopores is urgently needed. Here, we summarize the recent research on nanopore modification to achieve this goal. Modifying solid-state nanopores with different coating molecules can improve the selectivity, sensitivity, and stability of nanopores. The modified molecules can introduce different functions into the nanopores, greatly expanding the applications of this novel detection technology. We hope that this review of nanopore modification will provide new ideas for this field.

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