Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities

J. A. Carrillo, A. Jüngel, Peter Markowich, G. Toscani, A. Unterreiter

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195 Scopus citations

Abstract

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár-Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.

Original languageEnglish (US)
Pages (from-to)1-82
Number of pages82
JournalMonatshefte fur Mathematik
Volume133
Issue number1
DOIs
StatePublished - Jan 1 2001

Keywords

  • Asymptotic behavior
  • Degenerate diffusion
  • Degenerate parabolic systems
  • Differential inequalities
  • High-order diffusion
  • Rates of decay

ASJC Scopus subject areas

  • Mathematics(all)

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