A Global Regularity Result for the 2D Generalized Magneto-Micropolar Equations

Hui Zhang,Salah Mahmoud Boulaaras

doi:10.1155/2022/1501851

Abstract

In this paper, we proved the global (in time) regularity for smooth solution to the 2D generalized magneto-micropolar equations with zero viscosity. When there is no kinematic viscosity in the momentum equation, it is difficult to examine the bounds on the any derivatives of the velocity J ε u L 2 . In order to overcome the main obstacle, we find a new unknown quantity which is by combining the vorticity and the microrotation angular velocity; the structure of the system including the combined quantity obeys a Beale–Kato–Majda criterion. Moreover, the maximal regularity of parabolic equations together with the classic commutator estimates allows us to derive the H s estimates for solutions of the system.

Highlights

Introduction e standard3D incompressible magneto-micropolar equations can be written as ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨ ztu ztω + + u u · · ∇u − ∇ω −(μ + κ)Δu + cΔω − ]∇∇ ∇π ·ω + b · ∇b 4κω
Let J (I − Δ)1/2 denote the inhomogeneous differentiation operator; the proof of this lemma is based on some results due to Coifman and Meyer [23]
Combining with the above estimates and Gronwall’s inequality, we can obtain the desired global Hs bound

Summary

Introduction

1. Introduction e standard 3D incompressible magneto-micropolar equations can be written as E magneto-micropolar equations have been extensively studied and applied by many engineers and physicists [1]. Ortega-Torres and Rojas-Medar [3] established the local in time existence and uniqueness of strong solutions and proved global in time existence of strong solution for small initial data.

Results

Conclusion