Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

@article{Carrillo2011GlobalintimeWM,
  title={Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations},
  author={J. Carrillo and M. DiFrancesco and A. Figalli and T. Laurent and D. Slep{\vc}ev},
  journal={Duke Mathematical Journal},
  year={2011},
  volume={156},
  pages={229-271}
}
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 interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and… Expand
Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions
We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide aExpand
Measure solutions for non-local interaction PDEs with two species
This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems of non-local interaction PDEs with two species, which are the PDE counterpart of systems ofExpand
The Filippov characteristic flow for the aggregation equation with mildly singular potentials
Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity eldExpand
A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity
Abstract We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both theExpand
Weak solutions for one-dimensional aggregation equations
Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocityExpand
Weak solutions for one-dimensional aggregation equations March 22 , 2013
Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocityExpand
Stationary states of quadratic diffusion equations with long-range attraction
We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diusion arising in many contexts in population dynamics. The equation is theExpand
Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D
We prove the equivalence between the notion of Wasserstein gradient flow for a one- dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion ofExpand
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
Gradient flows for non-smooth interaction potentials
We deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolutionExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 60 REFERENCES
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-diffusionExpand
Large time behavior of nonlocal aggregation models with nonlinear diffusion
The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local Expand
Long-Time Asymptotics of Kinetic Models of Granular Flows
Abstract.We analyze the long-time asymptotics of certain one-dimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi-elastic limit of aExpand
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels !
We consider the multidimensional aggregation equation ut − ∇ (u∇K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). InExpand
Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2
We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, weExpand
A non-local model for a swarm
Abstract. This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists ofExpand
On an aggregation model with long and short range interactions
In recent papers the authors had proposed a stochastic model for swarm aggregation, based on individuals subject to long range attraction and short range repulsion, in addition to a classicalExpand
Uniqueness of Bounded Solutions to Aggregation Equations by Optimal Transport Methods
We show how to extend the method used in [22] to prove uniqueness of solutions to a family of several nonlocal equations containing aggregation terms and aggregation/diusion competition. They containExpand
Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak-Keller-Segel Model
TLDR
The convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated with the modified Patlak-Keller-Segel equation for subcritical masses is proved. Expand
FINITE-TIME SINGULARITIES OF AN AGGREGATION EQUATION IN R WITH FRACTIONAL DISSIPATION
We consider an aggregation equation in Rn, n ≥ 2, with fractional dissipation, namely, ut + ∇ · (u∇K ∗ u) = −ν(−∆)γ/2u , where 0 ≤ γ ≤ 2 and K is a nonnegative decreasing radial kernel with aExpand
...
1
2
3
4
5
...