Geometrical properties of solutions of the Porous Medium Equation for large times

Lee, Ki-Ahm; Vazquez, J. L.

doi:10.1512/iumj.2003.52.2200

Cited by 51 publications

(45 citation statements)

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“…We will mention two: eventual geometry (Lee and Vazquez (2003), [52]) and establishing convergence rates. and explain only the latter, since it motivates the work on nonlinear fractional diffusion.…”

Section: Nonlinear Central Limitmentioning

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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Section: Nonlinear Central Limitmentioning

confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

2012

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“…A time translationũ 0 (·) = u(·, 1) shows the exponent 1/t cannot be improved, and it agrees with the rate found by Dolbeault & del Pino and Otto at the transition point p = n. In the work announced below, we prove the sharp O(1/t) rate holds for the stronger norm (8) without the assumption of radial symmetry in the fastest range p ∈ ]0, 2] of conservative nonlinearities. The finite relative entropy hypothesis is replaced by (11)- (14). A result in this vein was conjectured by Carrillo & Vázquez, as well as Denzler & McCann [6].…”

Section: Main Textmentioning

confidence: 94%

“…Une translation en tempsũ 0 (·) = u(·, 1) démontre que la valeur de l'exposant 1/t est optimale ; c'est la même valeur que celle découverte par Dolbeault & del Pino et Otto au point de transition p = n. Dans les résultatsénoncés ci-dessous, nous obtenons la vitesse optimale O(1/t) au sens plus fort (8), sans hypothèse de symétrie, dans le régime le plus rapide p ∈ ]0, 2] des diffusions nonlinéaires conservatives. L'hypothèse sur l'entropie relative est remplacée par (11)- (14). Des résultats dans cette direction ont eté conjecturés par Carrillo & Vázquez, et aussi par Denzler & McCann [6].…”

Section: Version Française Abrégéeunclassified

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Sharp decay rates for the fastest conservative diffusions

Kim

McCann

2005

Comptes Rendus. Mathématique

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In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features -such as size and location -will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Here this rate is addressed for the fastest conservative nonlinearities in the singular diffusion equationwhich governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. A potential theoretic comparison technique is outlined below which establishes the sharp 1/t conjectured power law rate of decay uniformly in relative error, and in weaker norms such as L 1 (R n ).Dans les milieux dissipatifs, les perturbations initiales disparaissent progressivement, et seuls sont preservés leurs traits les plus grossiers, comme leur taille et leur position. Estimer précisément la vitesse de cette disparition est parfois une question d'un interêt primordial. Ici, nous donnons cette vitesse pour les diffusions nonlinéaires les plus rapides qui préservent la masse, pour le modèle (1) qui gouverne la diffusion d'une densité initiale, intégrable età support compact, vers un profil autosimilaire. Pour cela, nousétablissons une théorie de comparaison des potentiels, ce qui permet de montrer que la vitesse précise de décroissance est en 1/t pour la norme L 1 (R n ), et en fait uniforme pour l'erreur relative.The authors are grateful to José Antonio Carrillo and Dejan Slepčev for fruitful discussions, and to Agnes Tourin, Dario Cordero-Erausquin and Emmanuelle Servat for patient help with the version française. We thank the Fields Institute, and the Universities of Toronto, Marne-la-Vallée, and California at Los Angeles and Riverside, where parts of this work were performed, and many colleagues for the stimulating milieu which they helped to create.

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“…According to the different properties of the initial function u 0 (x), the corresponding nonnegative solutions may have different large time asymptotic behaviors, one can refer to the references [11][12][13][14][15][16][17]. In our article, we are going to study the large time asymptotic behavior for the solution of (1.1) and (1.2) by comparing it to the Barenblatt-type solution, let us give some details.…”

Section: Introductionmentioning

confidence: 99%

The asymptotic behavior of the solution of a doubly degenerate parabolic equation with the convection term

2012

J Inequal Appl

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The objective of this article is to study the large time asymptotic behavior of the nonnegative weak solution of the following nonlinear parabolic equationwith initial condition u(x, 0) = u 0 (x). By using Moser iteration technique, assuming that the uniqueness of the Barenblatt-type solution E c of the equation) is true, then the solution u may satisfywhich is uniformly true on the sets x ∈ R N : |x| < at) satisfies some growth order conditions, the exponents m and p satisfy m(p -1) >1.

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Geometrical properties of solutions of the Porous Medium Equation for large times

Cited by 51 publications

References 0 publications

Nonlinear Diffusion with Fractional Laplacian Operators

Nonlinear Diffusion with Fractional Laplacian Operators

Sharp decay rates for the fastest conservative diffusions

The asymptotic behavior of the solution of a doubly degenerate parabolic equation with the convection term

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