# Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion

## TL;DR

This paper investigates the long-term behavior of solutions to the 3D Hall-magneto-hydrodynamics system with minimal diffusion, showing conditions under which magnetic or velocity energies decay or stabilize over time.

## Contribution

It establishes the asymptotic energy behavior of solutions with only one diffusion term, demonstrating that even weak diffusion prevents energy oscillations.

## Key findings

- Magnetic energy vanishes in the resistive case with bounded velocity.
- Velocity energy vanishes in the viscous non-resistive case with bounded magnetic field.
- Energy converges to a constant in both cases under specified boundedness conditions.

## Abstract

This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space $\mathbb R^3$. We establish that, in the inviscid resistive case, the energy $\|b(t)\|_2^2$ vanishes and $\|u(t)\|_2^2$ converges to a constant as time tends to infinity provided the velocity is bounded in $W^{1-\alpha,\frac3\alpha}(\mathbb R^3)$; in the viscous non-resistive case, the energy $\|u(t)\|_2^2$ vanishes and $\|b(t)\|_2^2$ converges to a constant provided the magnetic field is bounded in $W^{1-\beta,\infty}(\mathbb R^3)$. In summary, one single diffusion, being as weak as $(-\Delta)^\alpha b$ or $(-\Delta)^\beta u$ with small enough $\alpha, \beta$, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.