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

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 ${6b(t)6}_2^2$ vanishes and ${6u(t)6}_2^2$ converges to a constant as time tends to infinity provided the velocity is bounded in $W^{1?\alpha ,\frac{3}{\alpha }}({\mathbb{R}}^3)$; in the viscous non-resistive case, the energy ${6u(t)6}_2^2$ vanishes and ${6b(t)6}_2^2$ 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 ${(?\mathrm{\Delta })}^{\alpha }b$ or ${(?\mathrm{\Delta })}^{\beta }u$ with small enough $\alpha ,\beta$, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.