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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 $u$ in this system is related to the vorticity $ω$ through the relations $u=\nabla^\perp ψ$ and $Δψ= Λ^σ(\log(I-Δ))^γω$, 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 $\nabla\times v =ω$ satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when $σ=0$ and $γ\ge 0$. In addition, the vorticity $ω$ is shown to be globally bounded in several functional settings such as $L^2$ for $σ>0$ in a suitable range.