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            <title xml:lang="en">Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities</title>
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            <funder>ANR NOnlinear methods in Nuclear and Atomic Physics" (NoNAP, ANR-10-BLAN-0101, 1/10/2010-30/9/2014)ANR Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations (STAB, ANR-12-BS01-0019, 2013-2016)ANR Kibord KInetic models in Biology Or Related Domains (ANR-13-BS01-0004, 1/1/2014-31/12/2017)</funder>
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                <term xml:lang="en">relative entropy</term>
                <term xml:lang="en">rates of convergence</term>
                <term xml:lang="en">porous medium equation</term>
                <term xml:lang="en">Ornstein-Uhlenbeck equation</term>
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                <term xml:lang="en">improved functional inequalities</term>
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                <term xml:lang="en">entropy -- entropy production inequalities</term>
                <term xml:lang="en">interpolation</term>
                <term xml:lang="en">Gagliardo-Nirenberg inequalities</term>
                <term xml:lang="en">generalized Poincaré inequalities</term>
                <term xml:lang="en">Gaussian isoperimetric inequality</term>
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              <p>This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincaré inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.</p>
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