Comptes Rendus
Partial Differential Equations
Sharp decay rates for the fastest conservative diffusions
Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 157-162.

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

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 L1(Rn).

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

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 L1(Rn), et en fait uniforme pour l'erreur relative.

Published online:
DOI: 10.1016/j.crma.2005.06.025

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
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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.

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