A uniform Besov boundedness and the well-posedness of the generalized dissipative quasi-geostrophic equation in the critical Besov space

Abstract

AbstractIn this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation∂t⁡θ+u⋅∇⁡θ+κ⁢Λ2⁢β⁢θ=0,(x,t)∈ℝ2×ℝ+,κ>0,\partial_{t}\theta+u\cdot\nabla\theta+\kappa\Lambda^{2\beta}\theta=0,\quad(x,t% )\in\mathbb{R}^{2}\times\mathbb{R}^{+},\;\kappa>0,whereu=-∇⊥⁡Λ-2+2⁢α⁢θ{u=-\nabla^{\bot}\Lambda^{-2+2\alpha}\theta},α∈[0,12]{\alpha\in[0,\frac{1}{2}]}andβ∈(0,1]{\beta\in(0,1]}. First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space.