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

In 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 ⁢ Λ 2 ⁢ β ⁢ θ = 0, (x, t) ∈ ℝ 2 × 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, where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ {u=-\nabla^{\bot}\Lambda^{-2+2\alpha}\theta}, α ∈ [ 0, 1 2 ] {\alpha\in[0,\frac{1}{2}]} and β ∈ (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.