Non-uniqueness of weak solutions for the fractal Burgers equation
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, p. 997-1016
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 -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.
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 . 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.
DOI : https://doi.org/10.1016/j.anihpc.2010.01.008
Classification:  35L65,  35L67,  35L82,  35S10,  35S30
Keywords: Fractional Laplacian, Non-local diffusion, Conservation law, Lévy–Khintchine's formula, Entropy solution, Admissibility of solutions, Oleĭnik's condition, Non-uniqueness of weak solutions
@article{AIHPC_2010__27_4_997_0,
     author = {Alibaud, Natha\"el and Andreianov, Boris},
     title = {Non-uniqueness of weak solutions for the fractal Burgers equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {4},
     year = {2010},
     pages = {997-1016},
     doi = {10.1016/j.anihpc.2010.01.008},
     zbl = {1201.35006},
     mrnumber = {2659155},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_4_997_0}
}
Alibaud, Nathaël; Andreianov, Boris. Non-uniqueness of weak solutions for the fractal Burgers equation. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, pp. 997-1016. doi : 10.1016/j.anihpc.2010.01.008. http://www.numdam.org/item/AIHPC_2010__27_4_997_0/

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