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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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