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This paper investigates the global (in time) regularity of solutions to asystem of equations that generalize the vorticity formulation of the 2DBoussinesq-Navier-Stokes equations. The velocity $u$ in this system is relatedto the vorticity $\omega$ through the relations $u=\nabla^\perp \psi$ and$\Delta \psi = \Lambda^\sigma (\log(I-\Delta))^\gamma \omega$, which reduces tothe 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 anequation of very special structure. This paper establishes the globalregularity and uniqueness of solutions for the case when $\sigma=0$ and$\gamma\ge 0$. In addition, the vorticity $\omega$ is shown to be globallybounded in several functional settings such as $L^2$ for $\sigma>0$ in asuitable range.

The 2D Boussinesq equations with logarithmically supercritical velocities