Hölder estimates for advection fractional-diffusion equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, p. 843-855

We analyse conditions for an evolution equation with a drift and fractional diffusion to have a Hölder continuous solution. In case the diffusion is of order one or more, we obtain Hölder estimates for the solution for any bounded drift. In the case when the diffusion is of order less than one, we require the drift to be a Hölder continuous vector field in order to obtain the same type of regularity result.

Published online : 2018-06-21
Classification:  35B65,  35R11
@article{ASNSP_2012_5_11_4_843_0,
author = {Silvestre, Luis},
title = {H\"older estimates for advection fractional-diffusion equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {4},
year = {2012},
pages = {843-855},
zbl = {1263.35056},
mrnumber = {3060702},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_843_0}
}

Silvestre, Luis. Hölder estimates for advection fractional-diffusion equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 843-855. http://www.numdam.org/item/ASNSP_2012_5_11_4_843_0/

[1] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 567–585. | Numdam | MR 2422079 | Zbl 1155.45004

[2] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (2007), 179–198. | MR 2283957 | Zbl 1129.47033

[3] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), 597–638. | MR 2494809 | Zbl 1170.45006

[4] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. 171 (2010), 1903–1930. | MR 2680400 | Zbl 1204.35063

[5] C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst. 27 (2010), 847–861. | MR 2600693 | Zbl 1194.35320

[6] P. Constantin, G. Iyer and J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 57 (2008), 2681–2692. | MR 2482996 | Zbl 1159.35059

[7] P. Constantin and J. Wu, Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 159–180. | Numdam | MR 2483817 | Zbl 1163.76010

[8] A. Kiselev and F. Nazarov Variation on a theme of Caffarelli and Vasseur, J. Math. Sci. 166 (2010), 31–39. | MR 2749211 | Zbl 1288.35393

[9] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace (English summary), Indiana Univ. Math. J. 55 (2006), 1155–1174. | MR 2244602 | Zbl 1101.45004

[10] L. Silvestre, Eventual regularization for the slightly supercritical quasi-geostrophic equation (English, French summary), Ann. Inst. H. Poincaré Anal. Non Linéare 27 (2010), 693–704. | Numdam | MR 2595196 | Zbl 1187.35186

[11] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Adv. in Math. 226 (2011), 2020–2039. | MR 2737806 | Zbl 1216.35165

[12] L. Silvestre, Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient, Discrete Contin. Dyn. Syst. 28 (2010), 1069–1081. | MR 2644779 | Zbl 1193.35240