Communications in Mathematical Sciences

Volume 22 (2024)

Number 6

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

Pages: 1529 – 1548

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n6.a4

Authors

Xueting Jin (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Quansen Jiu (School of Mathematical Sciences, Capital Normal University, Beijing, China)

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.

Keywords

two-dimensional micropolar equations, stability, Couette flow

2010 Mathematics Subject Classification

35B65, 35Q35, 76D03

Received 3 April 2023

Received revised 8 August 2023

Accepted 28 December 2023

Published 18 July 2024