Results 1  10
of
24
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
Abstract

Cited by 54 (19 self)
 Add to MetaCart
An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
Blowup in multidimensional aggregation equations with mildly singular interaction kernels
 Nonlinearity
, 2009
"... interaction kernels ..."
Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy Problem
"... We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity. It covers physically realistic mode ..."
Abstract

Cited by 25 (8 self)
 Add to MetaCart
We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity. It covers physically realistic models for granular materials. We prove (local in time) nonconcentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness is proved under additional smoothness assumption on the initial datum, for a restricted class of collision rates. Concerning the longtime behaviour, we give conditions for the cooling process to occur or not in finite time.
LongTime Asymptotics of Kinetic Models of Granular Flows
 Arch. Rational Mech. Anal
, 2003
"... We analyze the longtime asymptotics of certain onedimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coe#cient of restitution. These nonlinear ..."
Abstract

Cited by 21 (6 self)
 Add to MetaCart
We analyze the longtime asymptotics of certain onedimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coe#cient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending whether their similarity solutions (homogeneous cooling state) extinguish or not in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the similarity solution extinguishes in finite time, we prove that any other solution with initially bounded support extinguishes in finite time, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support.
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 properties of the inelastic Kac model
 J. Statist. Phys
, 2003
"... We introduce and discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distri ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
We introduce and discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distributed like stable laws, and attract initial densities which belong to the normal domain of attraction. If the initial density is assumed of finite energy, with higher moments bounded, it is shown that the solution converges for largetime to a profile with power law tails. These tails are heavily dependent on the collision rule.
The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels
 Chinese Annals of Mathematics, Series B
, 2009
"... with mildly singular interaction kernels ..."
Firstorder continuous models of opinion formation
 SIAM J. Appl. Math
, 2006
"... Abstract. We study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchange of opinion between individual agents. These models imply that the only possible final opinions are the extremal ones, and are similar to models of pure drift in magnetiza ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Abstract. We study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchange of opinion between individual agents. These models imply that the only possible final opinions are the extremal ones, and are similar to models of pure drift in magnetization. Both analytical and numerical methods allow to recover the final distribution of opinion between the two extremal ones. Key words. Nonlinear nonlocal hyperbolic equation, sociophysics, opinion formation, magnetization. AMS subject classifications. 91C20; 82B21; 60K35.
Asymptotics of blowup solutions for the aggregation equation
, 2011
"... We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an a ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behavior in the limits γ → 0 + and γ → 2 −.