Stability for the 2D Micropolar equations with partial dissipation near Couette flow

Abstract

In this paper, we will apply the Fourier multiplier method to explore the stability for the 2D micropolar equations with partial dissipation near Couette flow. The difficulty will be encountered due to the facts that one order derivative of the microtation appears on the right term of velocity equations and that the velocity equations only have vertical dissipation. To overcome the difficulty, we will make use of a Fourier multiplier to grasp the enhanced dissipation created by the special structure $y\partial_x-\nu\partial_{y}^2$ and obtain some new and higher-order estimates of the solution in an elegant way. Also, a time-dependent elliptic operator $\Lambda_t^b$ which commutes with linear part of the equations will be used to make our proof more clear.