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

José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel

Research output: Contribution to journalArticleResearchpeer-review

15 Citations (Scopus)

Abstract

In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus S1 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.
Original languageEnglish
Pages (from-to)1027-1050
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume6
Issue number5
Publication statusPublished - 1 Jan 2006

Keywords

  • Entropy
  • Entropy production
  • Entropy-entropy production method
  • Fast diffusion equation
  • Higher-order nonlinear PDEs
  • Logarithmic Sobolev inequality
  • Long-time behavior
  • Parabolic equations
  • Poincaré inequality
  • Porous media equation
  • Sobolev estimates

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