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New Liouville type theorems for the stationary MHD equations in $\mathbb{R}^3$

Wenke Tan

TL;DR

This work addresses Liouville-type questions for the 3D stationary MHD equations in $\mathbb{R}^3$ by shifting analysis to frequency space. It introduces a frequency-localized framework via Littlewood-Paley theory to connect the Dirichlet energy to low-frequency behavior, proving two new Liouville-type theorems for finite energy solutions. The first shows that a mild low-frequency bound on $u$ and $B$ in $L^\infty$-based Besov norms forces $u$ and $B$ to vanish, while the second establishes vanishing under a low-frequency $L^3$-norm condition on $u$. Together, these results highlight the leading role of the velocity field in the Liouville theory for stationary MHD and extend previous work by Chae-Weng.

Abstract

We research the Liouville type problem for the 3D stationary MHD equations in the frequency space. We establish two new Liouville type theorems for solutions with finite Dirichlet energy. Specifically, we show that the low-frequency part of the velocity field plays the leading role in a Liouville theory for MHD equations and then improve the results of Chae-Weng \cite{Chae-W}.

New Liouville type theorems for the stationary MHD equations in $\mathbb{R}^3$

TL;DR

This work addresses Liouville-type questions for the 3D stationary MHD equations in by shifting analysis to frequency space. It introduces a frequency-localized framework via Littlewood-Paley theory to connect the Dirichlet energy to low-frequency behavior, proving two new Liouville-type theorems for finite energy solutions. The first shows that a mild low-frequency bound on and in -based Besov norms forces and to vanish, while the second establishes vanishing under a low-frequency -norm condition on . Together, these results highlight the leading role of the velocity field in the Liouville theory for stationary MHD and extend previous work by Chae-Weng.

Abstract

We research the Liouville type problem for the 3D stationary MHD equations in the frequency space. We establish two new Liouville type theorems for solutions with finite Dirichlet energy. Specifically, we show that the low-frequency part of the velocity field plays the leading role in a Liouville theory for MHD equations and then improve the results of Chae-Weng \cite{Chae-W}.
Paper Structure (3 sections, 4 theorems, 95 equations)

This paper contains 3 sections, 4 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of results

Key Result

Theorem 1.1

Let $(u,B)$ be a weak solution of MHD in the class DI. Then $u=B=0$ if the following condition is valid Specifically, we deduce that if $(u,B)$ satisfies then $u=B=0$.

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 3.1