Entropy Dissipation Methods for Degenerate Parabolic Problems and Generalized Sobolev Inequalities

Content Provider	Semantic Scholar
Author	Carrillo, José A. Jüngel, Ansgar Markowich, Peter A. Toscani, Giuseppe Unterreiter, Andreas
Copyright Year	1999
Abstract	We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with con®nement by a uniformly convex potential, 2) uncon®ned scalar equations and 3) uncon®ned 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 CsiszaÂr±Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy. 2000 Mathematics Subject Classi®cation: 35B40, 35K55, 35K65, 35Q35, 35R45
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Alternate Webpage(s)	http://hera.ugr.es/doi/15088340.pdf
Language	English
Access Restriction	Open
Subject Keyword	Abnormal degeneration Arabic numeral 0 Bootstrapping Cations Convergence (action) Interpolation Kullback–Leibler divergence Naruto Shippuden: Clash of Ninja Revolution 3 Nash equilibrium Parabolic antenna Self-similarity Social inequality Solutions Uniformly convex space
Content Type	Text
Resource Type	Article