Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Haifeng Shang

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.

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

Abstract

This paper focuses on the global stability of the 3D magneto-micropolar equations with partial viscosity in the torus . 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.
Paper Structure (3 sections, 3 theorems, 110 equations)

This paper contains 3 sections, 3 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main1']}
  3. Proof of Theorem \ref{['main2']}

Key Result

Theorem 1.1

Consider the system MMEZ with $\chi>0$, $\kappa\geq0$, $\eta>0$ and $\nu>0$. Suppose that $(u_0, \omega_0, b_0) \in H^3(\mathbb T^3)$ with $\nabla\cdot u_0=\nabla\cdot b_0 = 0$ and $\int_{\mathbb T^3} u_0 dx=\int_{\mathbb T^3} \omega_0 dx=\int_{\mathbb T^3} b_0 dx=0$. Then there exists a constant $\ then system MMEZ has a unique global strong solution $(u, \omega, b)$ satisfying, for any $t > 0$ a

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • proof : Proof of Theorem \ref{['main1']}
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['main2']}