Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities

This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relativeentropies. When scales are taken into account and second moments xed accordingly, de cit functionals provide explicitstability measurements,i.e., bound with explicit constants distances to the manifold of optimal functions. Various resultsare obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalizedPoincar e inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in di usionequations (fast di usion, Ornstein-Uhlenbeck and porous medium equations) are obtained.nonnonouirechercheInternationa