Remarks on global well-posedness of mild solutions to the three-dimensional Boussinesq equations

Abstract

Recently, by using the argument of Lei & Lin (2011) [11], Liu & Gao (2017) [13] establish the global well-posedness of mild solutions to the three-dimensional Boussinesq equations in the space χ−1 defined by χ−1={u∈D′(R3):∫R3|ξ|−1|uˆ(ξ)|dξ<∞}. However, it seems that their proof is incorrect, and has some obvious and essential mistakes. Compared with the Navier-Stokes equations, it is difficulty to obtain a global well-posedness of mild solutions to the Boussinesq system in the space χ−1. In this paper, we will point out the mistakes of Liu & Gao. And, furthermore, in order to understand the difficulty of the Boussinesq system better, we study an illuminating system as follows:{∂tu+(u⋅∇)u−μ(1+t)αΔu+∇p=θe3,in R3×(0,∞),∂tθ+(u⋅∇)θ−k(1+t)αΔθ=0,in R3×(0,∞),∇⋅u=0,in R3×(0,∞),u(x,0)=u0,θ(x,0)=θ0,in R3, where μ>0, k>0 and α>1 are real constant parameters. By using the time-weighted estimate, we can show that the above system has a global mild solution.