Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities

Vincent Calvez (1, 2) , José Antonio Carrillo (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 UAB - Departament de Matemàtiques [Barcelona]

We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the logarithmic Hardy-Littlewood-Sobolev inequality in the one dimensional and radially symmetric two dimensional case based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect to Fourier distance in the subcritical case.

Mots clés: Keller-Segel; Long-time Asymptotics; Self-similarity; Wasserstein Distance; Contraction

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Calvez, José Antonio Carrillo. Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities. 2010. ⟨hal-00503203⟩