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Improved interpolation inequalities, relative entropy and fast diffusion equations

Abstract : We consider a family of Gagliardo-Nirenberg-Sobolev interpolation inequalities which interpolate between Sobolev's inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy - entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.
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https://hal.archives-ouvertes.fr/hal-00634852
Contributor : Jean Dolbeault Connect in order to contact the contributor
Submitted on : Wednesday, July 11, 2012 - 8:58:09 AM
Last modification on : Wednesday, October 14, 2020 - 4:01:31 AM
Long-term archiving on: : Friday, October 12, 2012 - 2:25:50 AM

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  • HAL Id : hal-00634852, version 2
  • ARXIV : 1110.5175

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Jean Dolbeault, Giuseppe Toscani. Improved interpolation inequalities, relative entropy and fast diffusion equations. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2013, 30 (5), pp.917-934. ⟨hal-00634852v2⟩

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