The 2D Boussinesq equations with logarithmically supercritical velocities

This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq–Navier–Stokes equations. The velocity uu in this system is related to the vorticity ωω through the relations u=∇⊥ψu=∇⊥ψ and Δψ=Λσ(log(I−Δ))γωΔψ=Λσ(log(I−Δ))γω, which reduces to the standard velocity–vorticity relation when σ=γ=0σ=γ=0. When either σ>0σ>0 or γ>0γ>0, the velocity uu is more singular. The “quasi-velocity” vv determined by ∇×v=ω∇×v=ω satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when σ=0σ=0 and γ≥0γ≥0. In addition, the vorticity ωω is shown to be globally bounded in several functional settings such as L2L2 for σ>0σ>0 in a suitable range.