Comptes Rendus
no. 11
Partial differential equations
On distributional solutions of local and nonlocal problems of porous medium type
[Sur des solutions distributionnelles de problèmes locaux et non locaux de type milieux poreux]

Présenté par : the Editorial Board

Félix del Teso1 ; Jørgen Endal1 ; Espen R. Jakobsen1
1 NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1154-1160.

Nous montrons l'unicité, l'existence, et des estimations a priori pour des solutions distributionnelles bornées de (0.1), où φ est continue et croissante et Lσ,μ est le générateur d'un processus de Lévy symétrique général. Cela veut dire que Lσ,μ peut avoir des parties locales et non locales, comme par exemple Lσ,μ=Δ−(−Δ)12. Nous présentons et montrons des nouveaux résultats d'unicité pour des solutions distributionnelles bornées de ce problème. Un nouveau résultat de type Liouville pour Lσ,μ joue un rôle clé. L'existence et des estimations a priori sont déduites d'une approximation numérique ; des inégalités de type énergie sont aussi obtenues.

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of

∂tu−Lσ,μ[φ(u)]=g(x,t)inRN×(0,T),(0.1)
where φ is merely continuous and nondecreasing, and Lσ,μ is the generator of a general symmetric Lévy process. This means that Lσ,μ can have both local and nonlocal parts like, e.g., Lσ,μ=Δ−(−Δ)12. New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for Lσ,μ. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.10.010
Félix del Teso 1 ; Jørgen Endal 1 ; Espen R. Jakobsen 1

1 NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 
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     title = {On distributional solutions of local and nonlocal problems of porous medium type},
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     pages = {1154--1160},
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Félix del Teso; Jørgen Endal; Espen R. Jakobsen. On distributional solutions of local and nonlocal problems of porous medium type. Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1154-1160. doi : 10.1016/j.crma.2017.10.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.010/

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[3] F. del Teso, J. Endal, E.R. Jakobsen, Numerical methods and analysis for nonlocal (and local) equations of porous medium type, preprint, 2017.

[4] F. del Teso; J. Endal; E.R. Jakobsen On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type, Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. A Volume in Honor of Helge Holden's 60th Birthday, EMS Ser. Congr. Rep., 2017 (in press)

[5] F. del Teso; J. Endal; E.R. Jakobsen Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type, Adv. Math., Volume 305 (2017), pp. 78-143

[6] J.L. Vázquez The Porous Medium Equation. Mathematical Theory, Oxf. Math. Monogr., Clarendon Press, Oxford University Press, Oxford, UK, 2007

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