Non-uniqueness of weak solutions for the fractal Burgers equation
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 soluti...
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| Veröffentlicht in: | Annales de l'Institut Henri Poincaré. Analyse non linéaire 2010-08, Vol.27 (4), p.997-1016 |
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| container_title | Annales de l'Institut Henri Poincaré. Analyse non linéaire |
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| creator | Alibaud, Nathaël Andreianov, Boris |
| description | 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_str_mv | 10.1016/j.anihpc.2010.01.008 |
| format | Article |
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| identifier | ISSN: 0294-1449 |
| ispartof | Annales de l'Institut Henri Poincaré. Analyse non linéaire, 2010-08, Vol.27 (4), p.997-1016 |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; European Mathematical Society Publishing House; NUMDAM |
| subjects | Admissibility of solutions Analysis of PDEs Conservation law Entropy solution Exact sciences and technology Fractional Laplacian Lévy–Khintchine's formula Mathematical analysis Mathematics Measure and integration Non-local diffusion Non-uniqueness of weak solutions Oleĭnik's condition Partial differential equations Sciences and techniques of general use |
| title | Non-uniqueness of weak solutions for the fractal Burgers equation |
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