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 . 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 -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, 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}, pages = {997--1016}, publisher = {Elsevier}, volume = {27}, number = {4}, year = {2010}, doi = {10.1016/j.anihpc.2010.01.008}, mrnumber = {2659155}, zbl = {1201.35006}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.008/} }
TY - JOUR AU - Alibaud, Nathaël AU - Andreianov, Boris TI - Non-uniqueness of weak solutions for the fractal Burgers equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 997 EP - 1016 VL - 27 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.008/ DO - 10.1016/j.anihpc.2010.01.008 LA - en ID - AIHPC_2010__27_4_997_0 ER -
%0 Journal Article %A Alibaud, Nathaël %A Andreianov, Boris %T Non-uniqueness of weak solutions for the fractal Burgers equation %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 997-1016 %V 27 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.008/ %R 10.1016/j.anihpc.2010.01.008 %G en %F 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, Tome 27 (2010) no. 4, pp. 997-1016. doi : 10.1016/j.anihpc.2010.01.008. http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.008/
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