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

@article{Carrillo2022FastDL,
  title={Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions},
  author={Jos{\'e} A. Carrillo and Matias G. Delgadino and Rupert L. Frank and Marica Lewin},
  journal={Mathematical Models and Methods in Applied Sciences},
  year={2022}
}
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… 

Figures from this paper

Some minimization problems for mean field models with competing forces

We review recent results on three families of minimization problems, defined on subsets of nonnegative functions with fixed integral. The competition between attractive and repulsive forces leads to

Reverse conformally invariant Sobolev inequalities on the sphere

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-sphere involving an operator of order 2s > n. In this case the Sobolev

Minimizers for a one-dimensional interaction energy

  • R. Frank
  • Mathematics
    Nonlinear Analysis
  • 2022

Infinite-time concentration in Aggregation–Diffusion equations with a given potential

References

SHOWING 1-10 OF 51 REFERENCES

Optimal critical mass in the two dimensional Keller–Segel model in R2

Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction

TLDR
The stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy is analyzed and it is proved that this global minimizers is a radially decreasing compactly supported con- tinuous density function which is smooth inside its support.

Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to

Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions

The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear

On Fast-Diffusion Equations with Infinite Equilibrium Entropy and Finite Equilibrium Mass

Abstract We extend the existing theory on large-time asymptotics for convection–diffusion equations, based on the entropy–entropy dissipation approach, to certain fast diffusion equations with

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation ut = Δum with m ∈ (0, 1) in the Euclidean space $${{\mathbb R}^d}$$, d ≧ 3, and study the asymptotic behavior of a natural class of

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations

Ground states in the diffusion-dominated regime

TLDR
The regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, is analyzed, and it is shown that all stationary states of the system are radially symmetric non-increasing and compactly supported.

Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates

The long-time asymptotics of certain nonlinear , nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4]
...