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Global Well-Posedness of a Non-local Burgers Equation: the periodic case
Cyril Imbert1; Roman Shvydkoy2; François Vigneron1
1 UPEC, LAMA, UMR 8050 du CNRS, 61 Avenue du Général de Gaulle, 94010 Creteil
2 Department of Mathematics, Stat. and Comp. Sci., M/C 249, University of Illinois, Chicago, IL 60607
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 4, pp. 723-758.

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads

ut-u|∇|u+|∇|(u2)=0.

We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in L ∞ . We show that any weak solution is instantaneously regularized into C ∞ . We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.

Cet article est consacré à l’étude d’une équation de Burgers non-locale, pour des données positives bornées et périodiques. Cette équation s’écrit :

ut-u|∇|u+|∇|(u2)=0.

Pour toute donnée positive régulière, nous construisons une unique solution globale classique. Pout toute donnée positive bornée, nous construisons une solution faible globale et nous démontrons que toute solution faible devient instantanément C ∞ . Nous décrivons aussi le comportement en temps long de toutes les solutions. Nos méthodes s’inspirent de plusieurs avancées récentes dans la théorie de la régularité parabolique des équations intégro-différentielles.

Published online:
DOI: 10.5802/afst.1509

Cyril Imbert 1; Roman Shvydkoy 2; François Vigneron 1

1 UPEC, LAMA, UMR 8050 du CNRS, 61 Avenue du Général de Gaulle, 94010 Creteil
2 Department of Mathematics, Stat. and Comp. Sci., M/C 249, University of Illinois, Chicago, IL 60607 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {723--758},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Cyril Imbert; Roman Shvydkoy; François Vigneron. Global Well-Posedness of a Non-local Burgers Equation: the periodic case. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 4, pp. 723-758. doi : 10.5802/afst.1509. https://afst.centre-mersenne.org/articles/10.5802/afst.1509/

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