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 (- 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 [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 L² to L^∞, from L^∞ 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^∞, 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 C^{1 - 2 α}. Higher regularity starting from C^δ with δ > 1 - 2 α 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].