Global well-posedness for a slightly supercritical surface quasi-geostrophic equation

We use a nonlocal maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field $u$ is obtained from the active scalar $\theta$ by a Fourier multiplier with symbol $i{k}^{\perp }\mathrm{\mid }k{\mathrm{\mid }}^{-1}m(k\mathrm{\mid })$, where $m$ is a smooth increasing function that grows slower than $\log \log \mathrm{\mid }k\mathrm{\mid }$ as $\mathrm{\mid }k\mathrm{\mid }\to \mathrm{\infty }$.