Entropy solution theory for fractional degenerate convection–diffusion equations
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 3, p. 413-441

We study a class of degenerate convection–diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection–diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.

⇒ corrected-by Erratum
DOI : https://doi.org/10.1016/j.anihpc.2011.02.006
Classification:  35R09,  35K65,  35A01,  35A02,  65M06,  65M12,  35B45,  35K59,  35D30,  35K57,  35R11
Keywords: Degenerate convection–diffusion equations, Fractional/fractal conservation laws, Entropy solutions, Uniqueness, Numerical method, Convergence
@article{AIHPC_2011__28_3_413_0,
author = {Cifani, Simone and Jakobsen, Espen R.},
title = {Entropy solution theory for fractional degenerate convection--diffusion equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {28},
number = {3},
year = {2011},
pages = {413-441},
doi = {10.1016/j.anihpc.2011.02.006},
zbl = {1217.35204},
mrnumber = {2795714},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2011__28_3_413_0}
}

Cifani, Simone; Jakobsen, Espen R. Entropy solution theory for fractional degenerate convection–diffusion equations. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 3, pp. 413-441. doi : 10.1016/j.anihpc.2011.02.006. http://www.numdam.org/item/AIHPC_2011__28_3_413_0/

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