Browse issues
no. 4
Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation
Cyril Imbert1; Tianling Jin2; Roman Shvydkoy3
1 Department of Mathematics and Applications, CNRS & École Normale Supérieure (Paris), 45 rue d’Ulm, 75005 Paris, France
2 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong and Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd., MS 305-16, Pasadena, CA 91125, USA
3 Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois, Chicago, IL 60607, USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 4, pp. 667-677.

We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in [6].

Nous obtenons des estimées de Schauder pour une classe générale d’équations linéaires intégro-différentielles. Ces estimées sont utilisées pour obtenir un résultat d’existence globale pour une équation scalaire de Burgers non-locale [6].

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1581

Cyril Imbert 1; Tianling Jin 2; Roman Shvydkoy 3

1 Department of Mathematics and Applications, CNRS & École Normale Supérieure (Paris), 45 rue d’Ulm, 75005 Paris, France
2 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong and Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd., MS 305-16, Pasadena, CA 91125, USA
3 Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois, Chicago, IL 60607, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2018_6_27_4_667_0,
     author = {Cyril Imbert and Tianling Jin and Roman Shvydkoy},
     title = {Schauder estimates for an integro-differential equation with applications to a nonlocal {Burgers} equation},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {667--677},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {4},
     year = {2018},
     doi = {10.5802/afst.1581},
     mrnumber = {3884608},
     zbl = {06984157},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1581/}
}
TY  - JOUR
AU  - Cyril Imbert
AU  - Tianling Jin
AU  - Roman Shvydkoy
TI  - Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
SP  - 667
EP  - 677
VL  - 27
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1581/
DO  - 10.5802/afst.1581
LA  - en
ID  - AFST_2018_6_27_4_667_0
ER  - 
%0 Journal Article
%A Cyril Imbert
%A Tianling Jin
%A Roman Shvydkoy
%T Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2018
%P 667-677
%V 27
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1581/
%R 10.5802/afst.1581
%G en
%F AFST_2018_6_27_4_667_0
Cyril Imbert; Tianling Jin; Roman Shvydkoy. Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 4, pp. 667-677. doi : 10.5802/afst.1581. https://afst.centre-mersenne.org/articles/10.5802/afst.1581/

[1] Luis Caffarelli; Chi Hin Chan; Alexis Vasseur Regularity theory for parabolic nonlinear integral operators, J. Am. Math. Soc., Volume 24 (2011) no. 3, pp. 849-869 | DOI | MR | Zbl

[2] Luis Caffarelli; Luis Silvestre Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math., Volume 62 (2009) no. 5, pp. 597-638 | DOI | MR | Zbl

[3] Hongjie Dong; Hong Zhang On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations (2016) (https://arxiv.org/abs/1604.00101) | Zbl

[4] Matthieu Felsinger; Moritz Kassmann Local regularity for parabolic nonlocal operators, Commun. Partial Differ. Equations, Volume 38 (2013) no. 9, pp. 1539-1573 | DOI | MR | Zbl

[5] Mariano Giaquinta; Enrico Giusti On the regularity of the minima of variational integrals, Acta Math., Volume 148 (1982), pp. 31-46 | DOI | MR | Zbl

[6] Cyril Imbert; Roman Shvydkoy; Francois Vigneron Global well-posedness of a non-local Burgers equation: The periodic case, Ann. Fac. Sci. Toulouse, Math., Volume 25 (2016) no. 4, pp. 723-758 | DOI | MR | Zbl

[7] Tianling Jin; Jingang Xiong Schauder estimates for solutions of linear parabolic integro-differential equations, Discrete Contin. Dyn. Syst., Volume 35 (2015) no. 12, pp. 5977-5998 | MR | Zbl

[8] Tianling Jin; Jingang Xiong Schauder estimates for nonlocal fully nonlinear equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 5, pp. 1375-1407 | MR | Zbl

[9] Héctor Chang Lara; Gonzalo Dàvila Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differ. Equ. (2014), pp. 1-2 | MR | Zbl

[10] Frédéric Lelièvre Approximation des equations de navier-stokes préservant le changement d’échelle, Université d’Évry Val d’Essonne (France) (2010) (Ph. D. Thesis)

[11] Frédéric Lelièvre A scaling and energy equality preserving approximation for the 3D Navier-Stokes equations in the finite energy case, Nonlinear Anal., Theory Methods Appl., Ser. A, Volume 74 (2011) no. 17, pp. 5902-5919 | DOI | MR | Zbl

[12] Remigijus Mikulevicius; Henrikas Pragarauskas On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., Volume 40 (2014) no. 4, pp. 539-563 | DOI | Zbl

Cited by Sources: