Sharp decay rates for the fastest conservative diffusions

Yong Jung Kim; Robert J. McCann

doi:10.1016/j.crma.2005.06.025

Partial Differential Equations

Sharp decay rates for the fastest conservative diffusions
[Vitesse de convergence optimale pour les diffusions nonlinéaires conservatives les plus rapides]

Présenté par : Haïm Brezis

Yong Jung Kim ¹ ; Robert J. McCann 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.

Résumé
Abstract

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.

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})

Reçu le : 2005-06-06
Publié le : 2005-08-01

DOI : 10.1016/j.crma.2005.06.025

Affiliations des auteurs :

Yong Jung Kim ¹ ; Robert J. McCann ²

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     title = {Sharp decay rates for the fastest conservative diffusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {157--162},
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     volume = {341},
     number = {3},
     year = {2005},
     doi = {10.1016/j.crma.2005.06.025},
     language = {en},
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Yong Jung Kim; Robert J. McCann. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.025/

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