Global Well-Posedness of a Non-local Burgers Equation: the periodic case
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 723-758.

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.

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.

@article{AFST_2016_6_25_4_723_0,
     author = {Imbert, Cyril and Shvydkoy, Roman and Vigneron, Fran\c{c}ois},
     title = {Global {Well-Posedness} of a {Non-local} {Burgers} {Equation:} the periodic case},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {723--758},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {4},
     year = {2016},
     doi = {10.5802/afst.1509},
     zbl = {1355.35190},
     mrnumber = {3564125},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1509/}
}
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Imbert, Cyril; Shvydkoy, Roman; Vigneron, François. Global Well-Posedness of a Non-local Burgers Equation: the periodic case. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 723-758. doi : 10.5802/afst.1509. http://archive.numdam.org/articles/10.5802/afst.1509/

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