Regularity criterion for the 3D Hall-magneto-hydrodynamics

TL;DR

This paper introduces a new, weaker regularity criterion for the 3D Hall-MHD system based on a wavenumber splitting approach inspired by turbulence theory, improving upon existing conditions.

Contribution

It proposes a novel regularity criterion for 3D Hall-MHD that is weaker than previous Prodi-Serrin type conditions, using a wavenumber splitting method.

Findings

The new criterion is weaker than existing ones.

It is based on energy flux estimates and turbulence-inspired ideas.

Provides a broader condition for regularity in 3D Hall-MHD.

Abstract

This paper studies the regularity problem for the 3D incompress- ible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi-Serrin type criteria) for the 3D Hall-MHD system.