Global Well-Posedness of a Non-local Burgers Equation: the Periodic Case

Cyril Imbert (1) , Roman Shvydkoy (2) , Francois Vigneron (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 UIC - University of Illinois [Chicago]

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads u_t − u|∇|u + |∇|(u^2) = 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.

Mots clés: parabolic integro-differential equations; Burgers; Euler; surface quasi-geostrophic equation; Schauder estimates; maximum principle

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Roman Shvydkoy, Francois Vigneron. Global Well-Posedness of a Non-local Burgers Equation: the Periodic Case. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2016, 25 (4), pp.723-758. ⟨hal-01160752⟩