Geodesic
A variation on a theme of Caffarelli and Vasseur
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 58-72 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur [1] showed that a certain class of weak solutions to the drift diffusion equation with initial data in $L^2$ gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the $C^\beta$ norm of the solution for some $\beta>0$. We use elementary tools involving control of Hölder norms using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasi-geostrophic (SQG) equation in addition to [5] and [1]. Bibl. – 6 titles. 
@article{ZNSL_2009_370_a3,
     author = {A. Kiselev and F. Nazarov},
     title = {A variation on a~theme of {Caffarelli} and {Vasseur}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {58--72},
     year = {2009},
     volume = {370},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a3/}
}
TY  - JOUR
AU  - A. Kiselev
AU  - F. Nazarov
TI  - A variation on a theme of Caffarelli and Vasseur
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2009
SP  - 58
EP  - 72
VL  - 370
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a3/
LA  - en
ID  - ZNSL_2009_370_a3
ER  - 
%0 Journal Article
%A A. Kiselev
%A F. Nazarov
%T A variation on a theme of Caffarelli and Vasseur
%J Zapiski Nauchnykh Seminarov POMI
%D 2009
%P 58-72
%V 370
%U http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a3/
%G en
%F ZNSL_2009_370_a3
A. Kiselev; F. Nazarov. A variation on a theme of Caffarelli and Vasseur. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 58-72. http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a3/

[1] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math/0608447 | MR

[2] P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, arXiv: math/0701592 | MR

[3] A. Cordoba, D. Cordoba, “A maximum principle applied to quasi-geostrophic equations”, Commun. Math. Phys., 249 (2004), 511–528 | DOI | MR | Zbl

[4] H. Dong, Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness, arXiv: math/0701826 | MR

[5] A. Kiselev, F. Nazarov, A. Volberg, “Global well-posedness for the critical $2D$ dissipative quasi-geostrophic equation”, Inv. Math., 167 (2007), 445–453 | DOI | MR | Zbl

[6] E. Stein, Harmonic Analysis, Princeton University Press, 1993 | MR | Zbl