Finite and infinite speed of propagation for porous medium equations with fractional pressure

Stan, Diana; del Teso, Félix; Vázquez, Juan Luis

doi:10.1016/j.crma.2013.12.003

Partial differential equations

Finite and infinite speed of propagation for porous medium equations with fractional pressure
[Vitesse de propagation finie et infinie pour des équations du milieu poreux avec une pression fractionnaire]

Présenté par : Haim Brezis

Stan, Diana ¹ ; del Teso, Félix ¹ ; Vázquez, Juan Luis ¹

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 123-128.

Résumé
Abstract

Nous étudions une équation du milieu poreux avec une pression potentielle fractionnaire : $\partial_{t} u = \nabla \cdot (u^{m - 1} \nabla p)$ , $p = {(- Δ)}^{- s} u$ , pour $m > 1$ , $0 < s < 1$ et $u (x, t) ⩾ 0$ . Le problème se pose pour $x \in R^{N}$ , $N ⩾ 1$ et $t > 0$ . La donnée initiale est supposée bornée avec support compact ou décroissance rapide à lʼinfini. Lorsque le paramètre m est variable, on obtient deux comportements différents comme suit : si $m \in [1, 2)$ , le problème a une vitesse de propagation infinie, alors que pour $m \in [2, \infty)$ , elle a une vitesse de propagation finie. On compare le résultat avec les comportements dʼautres modèles de diffusion non linéaire, qui sont très différents.

We study a porous medium equation with fractional potential pressure:

\partial_{t} u = \nabla \cdot (u^{m - 1} \nabla p), p = {(- Δ)}^{- s} u,

for

m > 1

0 < s < 1

and

u (x, t) ⩾ 0

. To be specific, the problem is posed for

x \in R^{N}

N ⩾ 1

, and

t > 0

. The initial data

u (x, 0)

is assumed to be a bounded function with compact support or fast decay at infinity. We establish the existence of a class of weak solutions for which we determine whether, depending on the parameter m, the property of compact support is conserved in time or not, starting from the result of finite propagation known for

m = 2

. We find that when

m \in [1, 2)

the problem has infinite speed of propagation, while for

m \in [2, \infty)

it has finite speed of propagation. Comparison is made with other nonlinear diffusion models where the results are widely different.

Reçu le : 2013-12-03
Accepté le : 2013-12-04
Publié le : 2014-01-01

DOI : 10.1016/j.crma.2013.12.003

Affiliations des auteurs :

Stan, Diana ¹ ; del Teso, Félix ¹ ; Vázquez, Juan Luis ¹

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

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     author = {Stan, Diana and del Teso, F\'elix and V\'azquez, Juan Luis},
     title = {Finite and infinite speed of propagation for porous medium equations with fractional pressure},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {123--128},
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     volume = {352},
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     doi = {10.1016/j.crma.2013.12.003},
     language = {en},
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Stan, Diana; del Teso, Félix; Vázquez, Juan Luis. Finite and infinite speed of propagation for porous medium equations with fractional pressure. Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 123-128. doi : 10.1016/j.crma.2013.12.003. http://www.numdam.org/articles/10.1016/j.crma.2013.12.003/

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