Global well-posedness for the micropolar fluid system in the critical Besov spaces

TL;DR

This paper establishes the global existence and uniqueness of solutions for the 3-D micropolar fluid system in critical Besov spaces, employing Fourier analysis and new $L^p$ estimates, and explores the solutions' long-term behavior.

Contribution

It introduces a novel approach combining solution transformations and Fourier localization to prove global well-posedness in critical Besov spaces for the micropolar fluid system.

Findings

Global well-posedness in critical Besov spaces.

Construction of solutions with highly oscillating initial data.

Decay estimates for long-term behavior of solutions.

Abstract

We prove the global well-posedness for the 3-D micropolar fluid system in the critical Besov spaces by making a suitable transformation to the solutions and using the Fourier localization method, especially combined with a new $L^p$ estimate for the Green matrix to the linear system of the transformed equation. This result allows to construct global solutions for a class of highly oscillating initial data of Cannone's type. Meanwhile, we analyze the long behavior of the solutions and get some decay estimates.