Global well-posedness for 3D incompressible magneto-micropolar fluids without resistivity and spin viscosity in strip domains

TL;DR

This paper proves the global existence and decay of classical solutions for a complex 3D incompressible magneto-micropolar fluid system without resistivity in a strip domain, overcoming significant analytical challenges.

Contribution

It extends the understanding of 3D incompressible magneto-micropolar fluids by establishing global well-posedness without resistivity and spin viscosity, using advanced energy methods.

Findings

Established global well-posedness of classical solutions.

Proved algebraic decay of solutions to equilibrium.

Overcame key analytical obstacles related to degeneracy and coupling.

Abstract

The global existence of classical solutions to the 3D compressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain was recently established by Feng, Hong, and Zhu [Sci. China Math., 2024]. While Lin and Xiang [Sci. China Math., 2020] established global well-posedness for the 2D incompressible counterpart, the global well-posedness for the 3D incompressible case remains open. The analysis is rendered difficult by three major obstacles which are further compounded in the 3D case: the degeneracy induced by the lack of magnetic diffusion and spin viscosity; the coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure; and the interaction between the velocity, magnetic field, and pressure, where the pressure acts as a non-state variable. In this paper, by adapting the two-layer energy method of Guo and Tice [Arch. Ration. Mech. Anal., 2013] and the techniques employed in Feng et al., together with refined trace estimates, we overcome these difficulties and establish the global well-posedness of classical solutions to the 3D incompressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain. Moreover, we demonstrate the algebraic time-decay of solutions toward the equilibrium.