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.

Abstract (VO)
Résumé

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.

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.

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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     author = {F\'elix del Teso and J{\o}rgen Endal and Espen R. Jakobsen},
     title = {On distributional solutions of local and nonlocal problems of porous medium type},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1154--1160},
     publisher = {Elsevier},
     volume = {355},
     number = {11},
     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/

Bibliographie
Cité par

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[2] A. de Pablo; F. Quirós; A. Rodríguez; J.L. Vázquez A general fractional porous medium equation, Commun. Pure Appl. Math., Volume 65 (2012) no. 9, pp. 1242-1284

[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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