Confinement for repulsive-attractive kernels

@article{Balague2012ConfinementFR,
  title={Confinement for repulsive-attractive kernels},
  author={D. Balagu'e and J. Carrillo and Y. Yao},
  journal={Discrete and Continuous Dynamical Systems-series B},
  year={2012},
  volume={19},
  pages={1227-1248}
}
We investigate the confinement properties of solutions of the aggregation equation with repulsive-attractive potentials. We show that solutions remain compactly supported in a large fixed ball depending on the initial data and the potential. The arguments apply to the functional setting of probability measures with mildly singular repulsive-attractive potentials and to the functional setting of smooth solutions with a potential being the sum of the Newtonian repulsion at the origin and a… Expand

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References

SHOWING 1-10 OF 53 REFERENCES
Nonlocal interactions by repulsive–attractive potentials: Radial ins/stability
Abstract We investigate nonlocal interaction equations with repulsive–attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulseExpand
Confinement in nonlocal interaction equations
Abstract We investigate some dynamical properties of nonlocal interaction equations. We consider sets of particles interacting pairwise via a potential W , as well as continuum descriptions of suchExpand
Equilibria of biological aggregations with nonlocal repulsive-attractive interactions
Abstract We consider the aggregation equation ρ t − ∇ ⋅ ( ρ ∇ K ∗ ρ ) = 0 in R n , where the interaction potential K incorporates short-range Newtonian repulsion and long-range power-law attraction.Expand
Dimensionality of Local Minimizers of the Interaction Energy
In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated with a repulsive–attractive potential. We show how the dimensionality of theExpand
Stability of stationary states of non-local equations with singular interaction potentials
TLDR
For locally attractive singular interaction potentials, which are singular in the sense that their first derivative is discontinuous at the origin, local non-linear stability of stationary states consisting of a finite sum of Dirac masses is proved. Expand
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations
In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interactingExpand
AGGREGATION VIA THE NEWTONIAN POTENTIAL AND AGGREGATION PATCHES
This paper considers the multidimensional active scalar problem of motion of a function ρ(x, t) by a velocity field obtained by v = −∇N ∗ρ, where N is the Newtonian potential. We prove well-posednessExpand
AGGREGATION AND SPREADING VIA THE NEWTONIAN POTENTIAL: THE DYNAMICS OF PATCH SOLUTIONS
This paper considers the multidimensional active scalar problem of motion of a function ρ(x, t) by a velocity field obtained by v = -∇N * ρ, where N is the Newtonian potential. We proveExpand
STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS
In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potentialExpand
Self-propelled particles with soft-core interactions: patterns, stability, and collapse.
TLDR
For the first time, a coherent theory is presented, based on fundamental statistical mechanics, for all possible phases of collective motion of driven particle systems, to predict stability and morphology of organization starting from the shape of the two-body interaction. Expand
...
1
2
3
4
5
...