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 ($\alpha <1/2$) dissipation $(-\mathrm{\Delta }{)}^{\alpha }$. 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 [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from $L^2$ to $L^{\infty }$, from $L^{\infty }$ to Hölder ($C^{\delta }$, $\delta >0$), 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 $C^{1-2\alpha }$. Higher regularity starting from $C^{\delta }$ with $\delta >1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].