Variation on a theme of caffarelli and vasseur
Tóm tắt
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.
Từ khóa
Tài liệu tham khảo
L. Caffarelli and A. Vasseur, “Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,” arXiv:math/0608447.
P. Constantin and J. Wu, “Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation,” arXiv:math/0701592.
A. Cordoba and D. Cordoba, “A maximum principle applied to quasi-geostrophic equations,” Commun. Math. Phys.,249, 511–528 (2004).
H. Dong, “Higher regularity for the critical and super-critical dissipative quasi-geostrophic equations,” arXiv:math/0701826.
A. Kiselev, F. Nazarov, and A. Volberg, “Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,” Inv. Math., 167, 445–453 (2007).
E. Stein, Harmonic Analysis, Princeton University Press (1993).