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

Abstract

We study a porous medium equation with fractional potential pressure: ∂ t u = ∇ · (u m−1 ∇p), p = (−∆) −s u, for m > 1, 0 < s < 1 and u(x, t) ≥ 0. To be specific, the problem is posed for x ∈ 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 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 ∈ [1, 2) the problem has infinite speed of propagation, while for m ∈ [2, ∞) it has finite speed of propagation. Comparison is made with other nonlinear diffusion models where the results are widely different. Résumé Vitesse de propagation finie et infinie pour deséquations du milieu poreux avec une pression fractionnaire. Nousétudions uneéquation du milieu poreux avec une pression potentielle fractionnaire: ∂ t u = ∇ · (u m−1 ∇p), p = (−∆) −s u, pour m > 1, 0 < s < 1 et u(x, t) ≥ 0. Le problème se pose pour x ∈ R N , N ≥ 1 et t > 0. La donnée initiale est supposée bornée avec support compact ou décroissance rapideà l'infini. Lorsque le paramètre m est variable, on obtient deux comportements différents comme suit: si m ∈ [1, 2) le problème a une vitesse de propagation infinie, alors que pour m ∈ [2, ∞), elle a une vitesse de propagation finie. On compare le résultat avec le comportement d'autres modèles de diffusion nonlinéaire qui est très différent.

Explicit Barenblatt profiles for fractional porous medium equations

“…Since both q and s are positive, the first equation is invalid and there is no Barenblatt profile of these equations. Therefore, there is no non-trivial Barenblatt profile of the type λ(R 2 − |x| 2 ) q + when m > 2, despite the existence of solutions propagating with finite speed in one dimension [16].…”

“…In contrast, the notable feature of (1.2) is the finite speed of propagation, studied for m = 2 by Caffarelli and Vázquez [7,8] and for general m > 1 by Biler, Imbert and Karch [3,4]. The variant (1.1b) has been studied only recently [16]; depending on m, the equation can have both finite (for 1 < m < 2) and infinite speed of propagation (for m > 2).…”

Porous medium equation with nonlocal pressure in a bounded domain

“…The same equation as in [18] appears in a one-dimensional model in dislocation theory that has also been studied by Biler et al [6]. Later mathematical works include [19,17,20], where regularity and asymptotic behaviour are established, paper [5] that treats the case m 1 = 1, m 2 > max{ 1−2s 1−s , 2s−1 N }, and the works [36,38,39] that treat the cases where m 1 = 1, and [37] that treats general exponents, see also [27].…”

“…• Question of finite speed of propagation, cf. works [37,38,39] for problems posed in the whole space. Regularity of free boundary problems, with open questions even for PME with nonlocal pressure, [18].…”

Initial-boundary value problem for a fractional type degenerate heat equation

“…A different approach to prove existence based on gradient flows has been developed by Lisini, Mainini and Segatti (see [20]). Then the model has been generalized in [26] [27] [28] [29] [30]. Uniqueness is still open in general, but under some truly restrictive regularity assumption is proven in [31].…”