Non-uniqueness of weak solutions for the fractal Burgers equation

Nathael Alibaud (1, 2) , Boris Andreïanov (2)

1 ENSMM - Ecole Nationale Supérieure de Mécanique et des Microtechniques
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the~$L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.

Mots clés: fractional laplacian; non-local diffusion; conservation law; Lévy-Khintchine's formula; entropy solution; admissibility of solutions; Oleinik condition; non-uniqueness of weak solutions

Domaines: Equations aux dérivées partielles [math.AP]

Nathael Alibaud, Boris Andreïanov. Non-uniqueness of weak solutions for the fractal Burgers equation. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2010, 27 (4), pp. 997-016. ⟨10.1016/j.anihpc.2010.01.008⟩. ⟨hal-00405892v3⟩