Linear and nonlinear enhanced dissipation for the 2-D micropolar equations near Couette flow

In this paper, we study the stability of two-dimensional micropolar equations perturbed around the Couette flow with small viscosity $ \nu>0 $. We first study the enhanced dissipation results for the linearized equations and show that the results are consistent with Navier–Stokes equations. Moreover, we study the nonlinear enhanced dissipation effect and prove that when the $ L^2 $ norm of the perturbation of the Couette flow is size at most $ \nu^{\frac{2}{3}} $, its size remains so for all times and the enhanced dissipation persists with a rate proportional to $ \nu^\frac{1}{3} $. The difficulty arises from the complex structure of the micropolar equations, specifically the fact that the vorticity equation involves a second-order derivative of the microrotation, whereas the vorticity in the microrotation equation has no derivative, leading to a mismatch in regularity. To overcome this, we fully utilize the structure of the system and use a Fourier multiplier method to complete the energy estimates and grasp enhanced dissipation.