Dislocation group dynamics III. similarity solutions of the continuum approximation

Head, A. K.

doi:10.1080/14786437208221020

Cited by 58 publications

(41 citation statements)

References 3 publications

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“…(CV) L'etude est complété dans les articles [5,6]. D'autre part, Head [12] a décrit la dynamique des dislocations dans les cristaux vues comme un continuum, et a proposé l'équation (1.1) avec s = 1/2, m = 2 si la dimension est N = 1. La densité de dislocations est de u = v x , donc v résout le "problème intégré"…”

Section: Version Française Abrégéeunclassified

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

Stan

Teso

Vázquez

2014

Comptes Rendus. Mathématique

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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.

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Section: Version Française Abrégéeunclassified

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

Stan

Teso

Vázquez

2014

Comptes Rendus. Mathématique

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“…There is a model for such dislocation phenomena proposed by A. K. Head [43] that leads to our equation in one space dimension with s = 1/2. It is written in an integrated version as…”

Section: Nonlocal Diffusion Model Of Porous Medium Typementioning

confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

2012

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We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that they describe the asymptotic behaviour of a wide class of solutions. The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1 -contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity. Nonlinear diffusion and fractional diffusionSince the work by Einstein [39] and Smoluchowski [62] at the beginning of the last century (cf. also Bachelier [9]), we possess an explantation of diffusion and Brownian motion in terms of the heat equation, and in particular of the Laplace operator. This explanation has had an enormous success both in Mathematics and Physics. In the decades that followed, the Laplace operator has been often replaced by more general types of so-called elliptic operators with variable coefficients, and later by nonlinear differential operators; a huge body of theory is now available, both for the evolution equations [49] and for the stationary states, described by elliptic equations of different kinds [50,42].

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“…In [18] existence of weak solutions is proved for initial data u 0 ∈ L 1 ∩L ∞ via an approximation method that requires a suitable decay of the data. On the other hand, the 1D model was investigated by Biler et al in [6] having an application to dislocation theory, [36]. Uniqueness is a key issue for this model.…”

Section: IImentioning

confidence: 99%

Porous Medium Equation with Nonlocal Pressure

Stan

Teso

Vázquez

2018

Current Research in Nonlinear Analysis

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We provide a rather complete description of the results obtained so far on the nonlinear diffusion equationwhich describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters m > 1 and 0 < s < 1, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when N = 1 and m > 2, and the asymptotic behavior of solutions when N = 1. The cases m = 1 and m = 2 were rather well known.Keywords: Nonlinear fractional diffusion, fractional Laplacian, existence of weak solutions, energy estimates, speed of propagation, smoothing effect, asymptotic behavior.

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Dislocation group dynamics III. similarity solutions of the continuum approximation

Cited by 58 publications

References 3 publications

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

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

Nonlinear Diffusion with Fractional Laplacian Operators

Porous Medium Equation with Nonlocal Pressure

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