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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 (α<1/2) dissipation (-Δ) α . 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 (α=1/2) QG equation [arXiv: math.AP/0608447 (2006)]. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from L 2 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 1-2α . Higher regularity starting from C δ with δ>1-2α 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].
MSC:
76D03Existence, uniqueness, and regularity theory
35Q35PDEs in connection with fluid mechanics
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