Sharp decay rates for the fastest conservative diffusions

Kim, Yong Jung; McCann, Robert J.

doi:10.1016/j.crma.2005.06.025

Partial Differential Equations

Sharp decay rates for the fastest conservative diffusions

Presented by: Haïm Brezis

Kim, Yong Jung ¹; McCann, Robert J.²

¹ Division of Applied Mathematics, KAIST, Gusong-dong 373-1, Yusong-gu, Taejon, 305-701 South Korea
² Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3, Canada

Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 157-162.

Abstract
Résumé

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 equation

u_{t} = Δ (u^{m}), {(n - 2)}_{+} / n < m ⩽ n / (n + 2), u, t ⩾ 0, x \in R^{n},

which 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

u_{t} = Δ (u^{m}), {(n - 2)}_{+} / n < m ⩽ n / (n + 2), u, t ⩾ 0, x \in R^{n},

qui gouverne la diffusion d'une densité initiale, intégrable et à support compact, vers un profil autosimilaire. Pour cela, nous établissons 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.

Received: 2005-06-06
Published online: 2005-08-01

DOI: 10.1016/j.crma.2005.06.025

Author's affiliations:

Kim, Yong Jung ¹; McCann, Robert J. ²

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Kim, Yong Jung; McCann, Robert J. Sharp decay rates for the fastest conservative diffusions. Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 157-162. doi : 10.1016/j.crma.2005.06.025. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.025/

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