Abstract

In this paper we study the decay characterization in the space HK+1σ(R3)×HKσ(R3) of solutions to a 3D magnetohydrodynamics-α model (MHD-α model for short) in the whole space R3, namely, ∥∥∇mu(t)∥∥2+α2∥∥∇m+1u(t)∥∥2+∥∥∇mB(t)∥∥2≤C(1+t)−min(r∗+m+32,m+52), where m≤K, r∗=min(r∗(u0),r∗(B0)) is the decay character of the initial datum (u0,B0)∈HK+1σ(R3)×HKσ(R3). We also get the optimal lower bounds for decay rates of solutions to the MHD-α model when −3/2<r∗≤1. The results obtained are extensions/improvements of previous results on decay rates of solutions to this 3D MHD-α model, the classical 3D MHD model, and the 3D viscous Camassa-Holm equations.