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Stability in Gagliardo-Nirenberg inequalities

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BDNS2020.pdf (604.1Kb)
Date
2020
Publishing date
07-2020
Collection title
Cahier de recherche CEREMADE
Link to item file
https://hal.archives-ouvertes.fr/hal-02887010
Dewey
Analyse
Sujet
Stability; Entropy methods; Harnack Principle; Gagliardo-Nirenberg inequality; Fast diffusion equation; Asymptotic behavior; Self-similar Barenblatt so-lutions; Rates of convergence; Spectral gap; Hardy-Poincaré inequalities; Intermediate asymptotics
URI
https://basepub.dauphine.fr/handle/123456789/21087
Collections
  • CEREMADE : Publications
Metadata
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Author
Bonforte, Matteo
82433 Departamento de Matemáticas
Dolbeault, Jean
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Nazaret, Bruno
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
92163 Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) [SAMM]
Simonov, Nikita
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Document de travail / Working paper
Item number of pages
52
Abstract (EN)
The purpose of this paper is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg inequalities. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. In both cases, the tail behaviour plays a key role. The regularity properties of the parabolic flow allow us to connect an improved entropy-entropy production inequality during the initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymp-totic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms the distance to the manifold of optimal functions.

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