Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. (English) Zbl 1163.76010
Summary: We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical dissipation . This study is motivated by a recent work of L. Caffarelli and A. Vasseur, in which they study the global regularity issue for the critical QG equation [arXiv: math.AP/0608447 (2006)]. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from to , from to Hölder , and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be , but it does not appear that their approach can be easily extended to establish the Hölder continuity of solutions. In order for their approach to work, we require the velocity to be in the Hölder space . Higher regularity starting from with can be established through Besov space techniques and will be presented elsewhere [P. Constantin and J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, in press].
|76D03||Existence, uniqueness, and regularity theory|
|35Q35||PDEs in connection with fluid mechanics|