On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation

TL;DR

This paper establishes the global existence, uniqueness, and decay rates of solutions for 2D micropolar equations with only angular velocity dissipation, addressing boundary and regularity challenges.

Contribution

It proves the global well-posedness and regularity of solutions under minimal assumptions, and analyzes their long-term decay behavior.

Findings

Global existence and uniqueness of solutions

Higher regularity with smoother initial data

Explicit decay rates for large-time behavior

Abstract

This paper focuses on the initial- and boundary-value problem for the two-dimensional micropolar equations with only angular velocity dissipation in a smooth bounded domain. The aim here is to establish the global existence and uniqueness of solutions by imposing natural boundary conditions and minimal regularity assumptions on the initial data. Besides, the global solution is shown to possess higher regularity when the initial datum is more regular. To obtain these results, we overcome two main difficulties, one due to the lack of full dissipation and one due to the boundary conditions. In addition to the global regularity problem, we also examine the large-time behavior of solutions and obtain explicit decay rates.