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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 $\left(\alpha <1/2\right)$ dissipation ${\left(-{\Delta }\right)}^{\alpha }$. 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 $\left(\alpha =1/2\right)$ 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 $\left({C}^{\delta },\delta >0\right)$, 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 35Q35 PDEs in connection with fluid mechanics