Some new Liouville type theorems for the 3D stationary magneto-micropolar fluid equations

TL;DR

This paper establishes new Liouville type theorems for 3D stationary magneto-micropolar fluid equations, relaxing growth conditions on solutions using innovative interpolation, energy, and ODE techniques.

Contribution

It introduces the most relaxed growth restrictions for angular velocity, allowing polynomial growth of its L^q-norm at infinity, and improves existing Liouville theorems.

Findings

Liouville theorems hold under polynomial growth of angular velocity.

Logarithmic factors relax growth conditions on velocity and magnetic fields.

Most relaxed restrictions are for angular velocity compared to velocity and magnetic fields.

Abstract

In this paper, we investigate Liouville type theorems for the 3D stationary magneto-micropolar fluid equations and micropolar fluid equations. Adopting an iteration procedure, taking advantage of the special structure of the equations and using a novel combination of interpolation techniques, we establish Liouville type theorems if the smooth solution satisfies certain growth conditions in terms of $L^p$-norms on the annuli. Furthermore, combining the energy method and some subtle ODE analysis, we relax the growth conditions on the velocity field and the magnetic field by logarithmic factors and obtain logarithmic improvement of Liouville type theorems. Compared with the velocity and the magnetic field, we raise the most relaxed restriction for the angular velocity. More specifically, we allow $L^q$-norm of the angular velocity on the annuli to grow polynomially at any degree, i.e. $\|\omega\|_{L^q(B_{2R}\backslash B_{3R/2})}$ is permitted to grow as fast as $R^N$ at infinity, where $N$ is an arbitrary positive integer.