Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

Abstract

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) 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 Hölder ($C^δ$, $δ>0$), and from Hölder 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 Hölder continuity of $L^\infty$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space $C^{1-2α}$. Higher regularity starting from $C^δ$ with $δ>1-2α$ can be established through Besov space techniques and will be presented elsewhere \cite{CW6}.