On the global well-posedness of a generalized 2D Boussinesq equations

Abstract

In this paper, we consider the global solutions to a generalized 2D Boussinesq equation \begin{align*} \left \{\begin{aligned} & \partial_{t} ω+ u\cdot \nabla ω+ νΛ^α ω= θ_{x_{1}} , \quad \\ & u = \nabla^{\bot} ψ= (-\partial_{x_{2}} , \partial_{x_{1}}) ψ, \quad Δψ= Λ^σ (\log (I-Δ))^γ ω, \quad \\ & \partial_{t} θ+ u\cdot \nabla θ+ κΛ^β θ= 0, \quad \\ & ω(x,0) = ω_{0}(x) , \quad θ(x,0) = θ_{0}(x), \end{aligned}\right. \end{align*} with $σ\geq 0$, $γ\geq 0$, $ν>0$, $κ>0$, $α< 1$ and $β< 1$. When $σ= 0$, $γ\geq 0$, $α\in [0.95,1)$ and $β\in (1-α,g(α))$, where $g(α)<1$ is an explicit function as a technical bound, we prove that the above equation has a global and unique solution in suitable functional space.