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Entropy-Energy inequalities and improved convergence rates for nonlinear parabolic equations

Juengel, Ansgar; Gentil, Ivan; Dolbeault, Jean; Carrillo, José A. (2006), Entropy-Energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems. Series B, 6, 5, p. 1027-1050. http://dx.doi.org/10.3934/dcdsb.2006.6.1027

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00008520/en/
Date
2006
Journal name
Discrete and Continuous Dynamical Systems. Series B
Volume
6
Number
5
Publisher
Southwest Missouri State University Dept. of Mathematics
Pages
1027-1050
Publication identifier
Metadata
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Author(s)
Juengel, Ansgar
Physik, Mathematik und Informatik [PMI]
Gentil, Ivan
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Carrillo, José A.
Abstract (EN)
In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus $S^1$ and explore their implications for the long-time asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropy-entropy production methods and based on appropriate Poincaré type inequalities.
Subjects / Keywords
long-time behavior; higher-order nonlinear PDEs; parabolic equations; Sobolev estimates; Poincaré inequality; Logarithmic Sobolev inequality; thin film equation; fast diffusion equation; porous media equation; entropy-entropy production method; entropy production; entropy