Saint-Venant Estimates and Liouville-Type Theorems for the Stationary MHD Equation in $\mathbb{R}^3$

TL;DR

This paper proves a Liouville-type theorem for stationary MHD equations in three dimensions, showing trivial solutions under weaker growth conditions by refining Saint-Venant estimates with Froullani integrals.

Contribution

It introduces a weaker growth condition for Liouville-type theorems in stationary MHD equations and refines Saint-Venant estimates using Froullani integrals.

Findings

Trivial solutions are characterized under weaker growth conditions.

Refined Saint-Venant estimates improve understanding of stationary MHD solutions.

The approach extends previous results by relaxing growth assumptions.

Abstract

In this paper, we investigate a Liouville-type theorem for the MHD equations using Saint-Venant type estimates. We show that \( (u, B) \) is a trivial solution if the growth of the \( L^s \) mean oscillation of the potential functions for both the velocity and magnetic fields are controlled. Our growth assumption is weaker than those previously known for similar results. The main idea is to refine the Saint-Venant type estimates using the Froullani integral.