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Stability in Gagliardo-Nirenberg inequalities

Abstract : 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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https://hal.archives-ouvertes.fr/hal-02887010
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Submitted on : Wednesday, July 1, 2020 - 10:06:59 PM
Last modification on : Monday, August 3, 2020 - 3:36:55 AM

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  • HAL Id : hal-02887010, version 1
  • ARXIV : 2007.03674

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Matteo Bonforte, Jean Dolbeault, Bruno Nazaret, Nikita Simonov. Stability in Gagliardo-Nirenberg inequalities. 2020. ⟨hal-02887010⟩

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