Global well-posedness for 3D compressible and incompressible micropolar fluids without angular viscosity in strip domains

TL;DR

This paper proves the global well-posedness of strong solutions for 3D micropolar fluids in strip domains, addressing both compressible and incompressible cases without angular viscosity, overcoming significant analytical challenges.

Contribution

It establishes the first global well-posedness results for 3D micropolar fluids with initial-boundary conditions in strip domains, including the compressible case, by exploiting system structure and energy estimates.

Findings

Proved global well-posedness of strong solutions near equilibrium.

Overcame difficulties caused by degeneracy and strong coupling.

Extended results to both compressible and incompressible micropolar fluids.

Abstract

This paper investigates an initial-boundary value problem for three-dimensional (3D) micropolar fluids in a strip domain, including both the compressible and the (homogeneous and inhomogeneous) incompressible cases in the absence of angular viscosity. The analysis is rendered difficult by two major obstacles: the degeneracy induced by vanishing angular viscosity, and the strong coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure. Moreover, the presence of physical boundaries in the strip domain further compounds these obstacles. While the global well-posedness of the 2D incompressible Cauchy problem has been established in the literature, no results are available for the 3D system and the initial-boundary value problem in both two and three dimensions, particularly in the compressible case. By exploiting the intrinsic structure of the system and establishing delicate energy estimates, we overcome these difficulties and prove the global well-posedness of strong solutions near equilibrium in a strip domain.