Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur have shown that a certain class of weak solutions to the drift diffusion equation with initial data in L2 gain H¨older continuity, provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on the BMO norm of a smooth velocity implies a uniform bound on the Cβ norm of the solution for some β > 0. We apply elementary tools involving the control of H¨older norms by using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasigeostrophic (SQG) equation in addition to the two proofs obtained earlier. Bibliography: 6 titles.
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References
L. Caffarelli and A. Vasseur, “Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,” arXiv:math/0608447.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 370, 2009, pp. 58–72.
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Kiselev, A., Nazarov, F. Variation on a theme of caffarelli and vasseur. J Math Sci 166, 31–39 (2010). https://doi.org/10.1007/s10958-010-9842-z
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DOI: https://doi.org/10.1007/s10958-010-9842-z