Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α < 1/2) dissipation (−∆). 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 [2]. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from L to L∞, from L∞ to Hölder (C, δ > 0), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be L∞, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L∞ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α. Higher regularity starting from C with δ > 1 − 2α can be established through Besov space techniques and will be presented elsewhere [10]. AMS (MOS) Numbers: 76D03, 35Q35