Results 1  10
of
50
Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations
"... Abstract. In this paper, we provide a wellposedness 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 pheno ..."
Abstract

Cited by 21 (9 self)
 Add to MetaCart
Abstract. In this paper, we provide a wellposedness 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 blowup time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have globalintime confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations. 1.
Asymptotic Flocking Dynamics for the kinetic CuckerSmale model
, 2009
"... Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmanntype equation. The largetime behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.
Probabilistic Approach for Granular Media Equations in the Non Uniformly Convex Case
, 2007
"... We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of CarrilloMcCannVil ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of CarrilloMcCannVillani [CMCV03, CMCV06] and completing results of Malrieu [Mal03] in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.
Functional inequalities, thick tails and asymptotics for the critical mass PatlakKellerSegel model, preprint
"... We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attractio ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R 2. We construct solutions of PatlakKellerSegel equation satisfying an entropyentropy dissipation inequality for Hλ. While the entropy dissipation for Hλ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropyentropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards λ. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp GagliardoNirenbergSobolev inequality.
RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION
"... Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these nforms represent two evolving distributions of particles over M, the minimum rootmeansquare distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be nonincreasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.
Some geometric calculations on Wasserstein space
, 2007
"... We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold. ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
Potential theory and optimal convergences rates in fast nonlinear diffusion. Preprint #23 at www.math.toronto.edu/∼mccann
"... A potential theoretic comparison technique is developed, which yields the conjectured optimal rate of convergence as t → ∞ for solutions of the fast diffusion equation ut = ∆(u m), (n − 2)+/n < m ≤ n/(n + 2), u, t ≥ 0, x ∈ R n, n ≥ 1 to a spreading selfsimilar profile, starting from integrable ini ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
A potential theoretic comparison technique is developed, which yields the conjectured optimal rate of convergence as t → ∞ for solutions of the fast diffusion equation ut = ∆(u m), (n − 2)+/n < m ≤ n/(n + 2), u, t ≥ 0, x ∈ R n, n ≥ 1 to a spreading selfsimilar profile, starting from integrable initial data with sufficiently small tails. This 1/t rate is achieved uniformly in relative error, and in weaker norms such as L 1 (R n). The range of permissible nonlinearities extends upwards towards m = 1 if the initial data shares enough of its moments with a specific selfsimilar profile. For example, in one space dimension, n = 1, the 1/t rate extends to the full range m ∈]0, 1 [ of nonlinearities provided the data is correctly centered. Résumé Dans les milieux dissipatifs, les perturbations initiales disparaissent progressivement, et seuls sont preservés leurs traits les plus grossiers, comme leur taille et leur position. Estimer précisément la vitesse de cette 〈 〈 disparition 〉〉 est parfois
A family of nonlinear fourth order equations of gradient flow type
, 2009
"... Global existence and longtime behavior of solutions to a family of nonlinear fourth order evolution equations on Rd are studied. These equations constitute gradient flows for the perturbed information functionals Fα,λ(u) = 1 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Global existence and longtime behavior of solutions to a family of nonlinear fourth order evolution equations on Rd are studied. These equations constitute gradient flows for the perturbed information functionals Fα,λ(u) = 1
On a nonlocal aggregation model with nonlinear diffusion
, 2008
"... Abstract. We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the wellposedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Abstract. We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the wellposedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops L ∞ xnorm blowup in finite time. 1. Introduction and
Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
, 2005
"... We investigate PrékopaLeindler type inequalities on a Riemannian manifold M equipped with a measure with density e−V where the potential V and the Ricci curvature satisfy Hessx V + Ricx ≥ λ I for all x ∈ M, with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M, ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
We investigate PrékopaLeindler type inequalities on a Riemannian manifold M equipped with a measure with density e−V where the potential V and the Ricci curvature satisfy Hessx V + Ricx ≥ λ I for all x ∈ M, with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the BakryEmery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.