Nonlinear diffusions, hypercontractivity and the optimal \(L^{p}\)-Euclidean logarithmic Sobolev inequality
type: Item
instance of: scholarly article
title: Nonlinear diffusions, hypercontractivity and the optimal \(L^{p}\)-Euclidean logarithmic Sobolev inequality
zbMATH Open document ID: 1058.35124
author: Jean Dolbeault
author: Ivan Gentil
author: Manuel del Pino
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-06T00:00:00Z
review text: The authors deal with the semigroup generated by the nonlinear diffusion equation \[ u_t= \Delta_p(u^{1/p-1})\tag{1} \] with \(\Delta_p w:= \text{div}(|\nabla w|^{p-2}\nabla w)\) for some \(p> 1\). They prove that associated to (1) semigroup is hyper- and ultra-contractive. To this end the authors use the \(L^p\)-Euclidean logarithmic Sobolev inequality. The last section of the paper is devoted to connections with large deviations and the Hamilton-Jacobi equation \[ v_t+{1\over p}|\nabla v|^p= 0 \] for which the optimal \(L^p\)-Euclidean logarithmic Sobolev inequality also plays an important role.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 47J35
zbMATH DE Number: 2082150
zbMATH Keywords: entropy
zbMATH Keywords: regularization
zbMATH Keywords: large deviations
zbMATH Keywords: Hamilton-Jacobi equations
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.009
OpenAlex ID: W2056211610
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DOI: 10.1016/J.JMAA.2003.10.009
description: scientific article; zbMATH DE number 2082150