The 2D incompressible Boussinesq equations with general critical dissipation

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This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to α+β=1 when Λα≡ (-Δ)α2 and Λβ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with α+β=1 and 0.9132≈ α₀<α<1. The cases when α=1 and when α=0 were previously resolved by Hmidi, Keraani and Rousset HKR1,HKR2. The global existence and uniqueness is achieved here by exploiting the global regularity of a generalized critical surface quasi-gesotrophic equation as well as the regularity of a combined quantity of the vorticity and the temperature.