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 R3. We establish that, in the inviscid resistive case, the energy ‖b(t)‖22 vanishes and ‖u(t)‖22 converges to a constant as time tends to infinity provided the velocity is bounded in W1−α,3α(R3); in the viscous non-resistive case, the energy ‖u(t)‖22 vanishes and ‖b(t)‖22 converges to a constant provided the magnetic field is bounded in W1−β,∞(R3). In summary, one single diffusion, being as weak as (−Δ)αb or (−Δ)βu with small enough α,β, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.