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

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.