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

Jing Loong, Guoxu Yang

TL;DR

This work establishes Liouville-type vanishing results for the stationary MHD equations in $\mathbb{R}^3$ under weakened growth conditions on the $L^s$ mean oscillations of velocity and magnetic-field potentials. By combining Saint-Venant-type estimates with the Froullani integral, the authors prove that if there exist smooth antisymmetric potentials $\boldsymbol{V},\boldsymbol{W}$ satisfying $\nabla\cdot\boldsymbol{V}=u$, $\nabla\cdot\boldsymbol{W}=B$ and for all $R>2$ one has $\|\boldsymbol{V}-(\boldsymbol{V})_{B_R}\|_{L^s(B_R)}+\|\boldsymbol{W}-(\boldsymbol{W})_{B_R}\|_{L^s(B_R)} \lesssim R^{\frac{s+6}{3s}}(\log R)^{\frac{s+6}{6s}}$ with $s\in(3,6]$, then the only smooth solution is $u\equiv B\equiv 0$. The approach improves previous results and clarifies the role of Saint-Venant-type and mean-oscillation techniques in proving Liouville properties for 3D stationary MHD, including a discussion of extensions to $s>6$ and connections to Navier–Stokes reductions.

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.

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

TL;DR

This work establishes Liouville-type vanishing results for the stationary MHD equations in under weakened growth conditions on the mean oscillations of velocity and magnetic-field potentials. By combining Saint-Venant-type estimates with the Froullani integral, the authors prove that if there exist smooth antisymmetric potentials satisfying , and for all one has with , then the only smooth solution is . The approach improves previous results and clarifies the role of Saint-Venant-type and mean-oscillation techniques in proving Liouville properties for 3D stationary MHD, including a discussion of extensions to and connections to Navier–Stokes reductions.

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 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.
Paper Structure (3 sections, 7 theorems, 61 equations)

This paper contains 3 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let ${u}$ and $B$ be smooth solutions of eq:MHD. Assume that there exists an $s \in(3,6]$ and smooth anti-symmetric potentials $\boldsymbol{V},\boldsymbol{W} \in C^{\infty}\left(\mathbb{R}^3 ; \mathbb{R}^{3 \times 3}\right)$ such that $\nabla \cdot \boldsymbol{V}=u$, $\nabla \cdot \boldsymbol{W}=B$, where $\beta = \frac{s+6}{6 s}$, for all $R>2$. Then $u=B \equiv 0$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • lemma 2.1: BY2024, Lemma 2.1
  • lemma 2.2: BY2024, Lemma 2.2
  • lemma 2.3: T2021, Lemma 3
  • ...and 6 more