The 2D Boussinesq equations with logarithmically supercritical velocities

AbstractThis 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 u in this system is related to the vorticity ω through the relations u=∇⊥ψ and Δψ=Λσ(log(I−Δ))γω, which reduces to the standard velocity–vorticity relation when σ=γ=0. When either σ>0 or γ>0, the velocity u is more singular. The “quasi-velocity” v determined by ∇×v=ω satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when σ=0 and γ≥0. In addition, the vorticity ω is shown to be globally bounded in several functional settings such as L2 for σ>0 in a suitable range