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 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 − α , 3 α (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 − β , ∞ (R 3). In summary, one single diffusion, being as weak as (− Δ) α b or (− Δ) β u with small enough α , β , is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system. [ABSTRACT FROM AUTHOR]