The 2D regularized incompressible Boussinesq equations with general critical dissipations

Considering the 2D regularized Boussinesq equations with fractional dissipations (Λαu,Λβθ) and convection terms (Λ−γu⋅∇u,Λ−γu⋅∇θ), where Λ=−Δ and γ≥0, we prove the global existence and uniqueness of the solution in two critical cases. The first case has fractional dissipations (Λαu,Λβθ), where α+β=1−γ,β>0, and the second one has particular dissipation (Λ1−γu,0). In particular, for the case γ=0, we give some decay estimates for (θ,u) and the uniform estimate for G independent of time, where G=∂1u2−∂2u1−∂1Λ−αθ.