Global well-posedness and large-time decay for the 2D micropolar equations

Abstract

This paper studies the global (in time) regularity and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. When there is no kinematic viscosity in the momentum equation, the global regularity problem is not easy due to the lack of suitable bounds on the derivatives. The idea here is to fully exploit the structure of the system and control the vorticity via the evolution equation of a combined quantity of the vorticity and the micro-rotation angular velocity. To understand the large time behavior, we overcome two main difficulties, the lack of kinematic viscosity and the presence of linear terms. Classical tools such as the Fourier splitting method of Schonbek and Kato's approach for the decay of small solutions do not apply here. We introduce a diagonalization process to eliminate the linear terms and rely on the uniform bounds for the first derivatives of the solutions to generate suitable decay rates.