Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions

José A. Carrillo (1) , Matias G. Delgadino (1, 2) , Rupert L. Frank (3, 4) , Mathieu Lewin (5, 6)

1 MI - Mathematical Institute [Oxford]
2 PUC-Rio - Pontifícia Universidade Católica do Rio de Janeiro [Brasil] = Pontifical Catholic University of Rio de Janeiro [Brazil] = Université catholique pontificale de Rio de Janeiro [Brésil]
3 CALTECH - California Institute of Technology
4 LMU - Mathematisches Institut [München]
5 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
6 PSL - Université Paris Sciences et Lettres

We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions $N\geq6$. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions $N\geq3$, for homogeneous interaction potentials with higher power.

Domaines: Physique mathématique [math-ph]

José A. Carrillo, Matias G. Delgadino, Rupert L. Frank, Mathieu Lewin. Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions. Mathematical Models and Methods in Applied Sciences, In press, 32 (4), pp.831-850. ⟨10.1142/S021820252250018X⟩. ⟨hal-03079032⟩