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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 \((\alpha < 1/2)\) dissipation \((-\Delta )^{\alpha }\). 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 \((\alpha = 1/2)\) QG equation [arXiv: math.AP/0608447 (2006)]. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from \(L^{2}\) to \(L^{\infty }\), from \(L^{\infty }\) to Hölder \((C^{\delta }, \delta > 0)\), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be \(L^{\infty }\), but it does not appear that their approach can be easily extended to establish the Hölder continuity of \(L^{\infty }\) solutions. In order for their approach to work, we require the velocity to be in the Hölder space \(C^{1-2\alpha }\). Higher regularity starting from \(C^{\delta }\) with \(\delta > 1 - 2\alpha \) 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:
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
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