Holder 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 [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-infinity, from L-infinity to Holder (C-delta, delta > 0), and from Holder to classical Solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be L-infinity, but it does not appear that their approach can be easily extended to establish the Holder continuity of L-infinity solutions. In order for their approach to work, we require the velocity to be in the Holder space C1-2 alpha. Higher regularity starting from C with > 1-2 alpha can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Holder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincare Anal. Non Lineaire, in press]. (c) 2007 Elsevier Masson SAS. All rights reserved.