The stabilizing effect of the microstructure on the 3D magneto-micropolar equations

TL;DR

This paper demonstrates that microstructure in 3D magneto-micropolar equations enhances dissipation and stability, providing new insights into the stabilization of electrically conducting fluids with partial viscosity.

Contribution

It establishes the first results showing the stabilizing effect of microstructure on the 3D magneto-micropolar equations with partial viscosity.

Findings

Proves global stability and exponential decay for equations with zero kinematic viscosity.

Shows algebraic decay near a background magnetic field with zero magnetic diffusion.

Highlights the microstructure's role in stabilizing the fluid by enhancing dissipation.

Abstract

This paper focuses on the global stability of the 3D magneto-micropolar equations with partial viscosity in the torus $\mathbb T^3$. We first establish the global stability and exponential decay for the 3D magneto-micropolar equations with zero kinematic viscosity. If the micro-rotation effect is neglected, this system reduces to the 3D inviscid and resistive MHD equations which stability problem is still a challenging open problem. Secondly, we obtain the global stability and algebraic decay to the 3D magneto-micropolar equations with zero kinematic viscosity and zero magnetic diffusion on perturbations near a background magnetic field. This system becomes the 3D ideal MHD equations by ignoring the microstructure, and it is well-known that the weighted spaces must be introduced to show the global well-posedness of the ideal MHD equations. Our results indicate that the microstructure has the effect of enhancing dissipation and contributes to stabilize the fluid. To the best of our knowledge, these are the first results on the stabilizing effect of the microstructure on electrically conducting fluids.