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 $\sigma \ge 0$, $\gamma \ge 0$, $\nu >0$, $\kappa >0$, $\alpha <1$ and $\beta <1$. When $\sigma =0$, $\gamma \ge 0$, $\alpha \in [0.95,1)$ and $\beta \in (1-\alpha ,g(\alpha ))$, where $g(\alpha )<1$ is an explicit function as a technical bound, we prove that the above equation has a global and unique solution in suitable functional space.

Submitted to NODEA about one year ago. arXiv admin note: text overlap with arXiv:0910.0311, arXiv:1111.2082 by other authors

References in corpus (1)

• Global well-posedness of the critical Burgers equation in critical Besov spaces

Cited by in corpus (1)

• A global regularity result for the 2D Boussinesq equations with critical dissipation