October 3, 2025Open Access

Liouville type theorem for double Beltrami solutions of the Hall-MHD system in R³

Key Points

The theorem confirms that if double Beltrami solutions exist under specific integral conditions, both the velocity and magnetic field must be zero.
Specifically, for double Beltrami solutions in Hall-MHD, the stipulation of finite integrals leads to the conclusion that both components vanish.
This finding extends previous results regarding Beltrami solutions for Euler equations, showcasing a relation between these systems.
The proof relies on conditions that ensure the smoothness and behavior of solutions in the context of the stationary Hall-MHD equations.

Abstract

In this paper we prove Liouville type theorem for the double Beltrami solutions to the stationary Hall-MHD equations in R³. Let (u, B) be a smooth double Beltrami solution to the stationary Hall-MHD equations in R³, satisfying ₑ℃ (|u|q + |B|q) dx <+ for some q [2, 3), then u=B=0. In the case of B=0 the theorem reduces the previously known Liouville type result for the Beltrami solutions to the Euler equations.

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Cite This Study

Dongho Chae (Thu,) studied this question.

synapsesocial.com/papers/68e03501f0e39f13e7fa38ba https://doi.org/https://doi.org/10.48550/arxiv.2505.00885

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