Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization

Jean Dolbeault (1) , Maria J. Esteban (1) , Michael Loss (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 School of Mathematics - Georgia Institute of Technology

This paper is devoted to the computation of the asymptotic boundary terms in entropy methods applied to a fast diffusion equation with weights associated with Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, only elliptic equations have been considered and our goal is to justify, at least partially, an extension of the carré du champ / Bakry-Emery / Rényi entropy methods to parabolic equations. This makes sense because evolution equations are at the core of the heuristics of the method even when only elliptic equations are considered, but this also raises difficult questions on the regularity and on the growth of the solutions in presence of weights. We also investigate the relations between the optimal constant in the entropy - entropy production inequality, the optimal constant in the information - information production inequality, the asymptotic growth rate of generalized Rényi entropy powers under the action of the evolution equation and the optimal range of parameters for symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities, under the assumption that the weights do not introduce singular boundary terms at x=0. These considerations are new even in the case without weights. For instance, we establish the equivalence of carré du champ and Rényi entropy methods and explain why entropy methods produce optimal constants in entropy - entropy production and Gagliardo-Nirenberg inequalities in absence of weights, or optimal symmetry ranges when weights are present.

Mots clés: Caffarelli-Kohn-Nirenberg inequalities; Gagliardo-Nirenberg inequalities; weights; optimal functions; symmetry; symmetry breaking; optimal constants; improved inequalities; parabolic flows; fast diffusion equation; self-similar solutions; asymptotic behavior; intermediate asymptotics; rate of convergence; entropy methods; carré du champ; Rényi entropy powers; entropy - entropy production inequality; self-similar variables; bifurcation; instability; rigidity results; linearization; spectral estimates; spectral gap; Hardy-Poincaré inequality

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban, Michael Loss. Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization. Journal of Elliptic and Parabolic Equations, 2016, 2, pp.267-295. ⟨hal-01395771⟩