On distributional solutions of local and nonlocal problems of porous medium type

Félix del Teso; Jørgen Endal; Espen R. Jakobsen

doi:10.1016/j.crma.2017.10.010

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 Teso ¹ ; Jørgen Endal ¹ ; Espen R. Jakobsen ¹

¹ NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1154-1160.

Résumé
Abstract

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 $L^{σ, μ}$ est le générateur d'un processus de Lévy symétrique général. Cela veut dire que $L^{σ, μ}$ peut avoir des parties locales et non locales, comme par exemple $L^{σ, μ} = Δ - {(- Δ)}^{\frac{1}{2}}$ . Nous présentons et montrons des nouveaux résultats d'unicité pour des solutions distributionnelles bornées de ce problème. Un nouveau résultat de type Liouville pour $L^{σ, μ}$ 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

\partial_{t} u - L^{σ, μ} [φ (u)] = g (x, t) in R^{N} \times (0, T),

(0.1)

where φ is merely continuous and nondecreasing, and

L^{σ, μ}

is the generator of a general symmetric Lévy process. This means that

L^{σ, μ}

can have both local and nonlocal parts like, e.g.,

L^{σ, μ} = Δ - {(- Δ)}^{\frac{1}{2}}

. 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

L^{σ, μ}

. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.

Reçu le : 2017-06-16
Accepté le : 2017-10-17
Publié le : 2017-11-01

DOI : 10.1016/j.crma.2017.10.010

Affiliations des auteurs :

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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     year = {2017},
     doi = {10.1016/j.crma.2017.10.010},
     language = {en},
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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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