Non-uniqueness of weak solutions for the fractal Burgers equation

Alibaud, Nathaël; Andreianov, Boris

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 4, pp. 997-1016.

La notion de solution entropique de Kruzhkov a été étendue par Alibaud en 2007 aux lois de conservation avec un terme diffusif fractionnaire ; ceci a permis de démontrer que le prolème de Cauchy est bien posé dans le cadre $L^{\infty }$. Dans cet article, on montre que si l'ordre de l'opérateur de diffusion est strictement plus petit que un, alors il peut exister plusieurs solutions faibles ; on apporte ainsi une motivation supplémentaire à l'utilisation des solutions entropiques.

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.

Classification : 35L65, 35L67, 35L82, 35S10, 35S30
Mots clés : Fractional Laplacian, Non-local diffusion, Conservation law, Lévy–Khintchine's formula, Entropy solution, Admissibility of solutions, Oleĭnik's condition, Non-uniqueness of weak solutions

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