Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities

Carillo, José A.; Jüngel, Ansgar; Markovich, Peter A.; Toscani, Giuseppe; Unterreiter, Andreas

Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities

Dateien

preprint_105.pdfGröße: 776.43 KBDownloads: 691

Datum

1999

Autor:innen

Carillo, José A.

Jüngel, Ansgar

Markovich, Peter A.

Toscani, Giuseppe

Unterreiter, Andreas

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Informatik und Informationswissenschaft

Publikationstyp

Preprint

Publikationsstatus

Published

Zusammenfassung

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 Csizar-Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.

Fachgebiet (DDC)

004 Informatik

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ISO 690

CARILLO, José A., Ansgar JÜNGEL, Peter A. MARKOVICH, Giuseppe TOSCANI, Andreas UNTERREITER, 1999. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities

BibTex

@unpublished{Carillo1999Entro-6436,
  year={1999},
  title={Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities},
  author={Carillo, José A. and Jüngel, Ansgar and Markovich, Peter A. and Toscani, Giuseppe and Unterreiter, Andreas}
}

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    <dcterms:abstract xml:lang="eng">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&lt;br /&gt;approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality.&lt;br /&gt;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 Csizar-Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.</dcterms:abstract>
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