Time decay rate of weak solutions to the generalized MHD equations in R2R2

This paper studies the time decay rate of weak solutions to the following two-dimensional magnetohydrodynamics (MHD) equations with fractional dissipations ∂tu+(u·∇)u−(b·∇)b+∇p=−(−▵)αu,∂tb+(u·∇)b−(b·∇)u=−(−▵)βb.The motivation is to understand how the parameters α and β affect the decay rate of its solutions. The authors use the Fourier splitting method of Schonbek to prove that the solutions have the following decay rate ∥u(x,t)∥2+∥b(x,t)∥2⩽c(1+t)1−2/γ,forlargeenought,where α, β   ∈ [1, 2) and γ=max{α,β}γ=max{α,β}.