H\"older continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($\alpha <1/2$) dissipation $(-\Delta)^\alpha$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($\alpha = 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to H\"{o}lder ($C^{\delta}$, $\delta>0$), and from H\"{o}lder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it does not appear that their approach can be easily extended to establish the H\"{o}lder continuity of $L^\infty$ solutions. In order for their approach to work, we require the velocity to be in the H\"{o}lder space $C^{1-2\alpha}$. Higher regularity starting from $C^\delta$ with $\delta>1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere \cite{CW6}.