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 $\omega$ through the relations $u=\nabla^\perp \psi$ and $\Delta \psi = \Lambda^\sigma (\log(I-\Delta))^\gamma \omega$, which reduces to the standard velocity-vorticity relation when $\sigma=\gamma=0$. When either $\sigma>0$ or $\gamma>0$, the velocity $u$ is more singular. The "quasi-velocity" $v$ determined by $\nabla\times v =\omega$ satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when $\sigma=0$ and $\gamma\ge 0$. In addition, the vorticity $\omega$ is shown to be globally bounded in several functional settings such as $L^2$ for $\sigma>0$ in a suitable range.