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This paper studies the asymptotic behavior of smooth solutions to thegeneralized Hall-magneto-hydrodynamics system (1.1) with one single diffusionon the whole space $\mathbb R^3$. We establish that, in the inviscid resistivecase, the energy $\|b(t)\|_2^2$ vanishes and $\|u(t)\|_2^2$ converges to aconstant 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 constantprovided the magnetic field is bounded in $W^{1-\beta,\infty}(\mathbb R^3)$. Insummary, one single diffusion, being as weak as $(-\Delta)^\alpha b$ or$(-\Delta)^\beta u$ with small enough $\alpha, \beta$, is sufficient to preventasymptotic energy oscillations for certain smooth solutions to the system.

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