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277
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 ..."
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Cited by 54 (19 self)
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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.
Optimal mass transportation and Mather theory
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 2005
"... We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz ..."
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Cited by 33 (4 self)
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.
Large deviations for stochastic processes
, 2000
"... weakly interacting particles, viscosity solutions, comparison principle, mass transport techniques ..."
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Cited by 31 (2 self)
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weakly interacting particles, viscosity solutions, comparison principle, mass transport techniques
On the second boundary value problem for MongeAmpère type equations and optimal
"... Abstract. This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for MongeAmpère type equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of the condition A3, which was int ..."
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Cited by 31 (1 self)
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Abstract. This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for MongeAmpère type equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of the condition A3, which was introduced in a recent paper with Xinan Ma in conjunction with interior regularity. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions. 1.
Blowup in multidimensional aggregation equations with mildly singular interaction kernels
 Nonlinearity
, 2009
"... interaction kernels ..."
Transportation costinformation inequalities and applications to random dynamical systems and diffusions
 Ann. Probab
"... We first give a characterization of the L 1transportation costinformation inequality on a metric space and next find some appropriate sufficient condition to transportation costinformation inequalities for dependent sequences. Applications to random dynamical systems and diffusions are studied. 1. ..."
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Cited by 27 (5 self)
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We first give a characterization of the L 1transportation costinformation inequality on a metric space and next find some appropriate sufficient condition to transportation costinformation inequalities for dependent sequences. Applications to random dynamical systems and diffusions are studied. 1. Introduction and questions. Let (E,d
Exponential decay for the fragmentation or celldivision equation
 J. Diff. Eq
, 2003
"... We consider a classical integrodifferential equation that arises in various applications as a model for celldivision or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady dynamics (first positive eigenvector ..."
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Cited by 25 (7 self)
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We consider a classical integrodifferential equation that arises in various applications as a model for celldivision or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady dynamics (first positive eigenvector) under general assumptions in the variable coefficients case. We also prove the exponential convergence, for large times, of solutions toward such a steady state.
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 ..."
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Cited by 20 (5 self)
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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.
Global solutions of the HunterSaxton equation
 SIAM J. Math. Anal
"... Abstract. We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respec ..."
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Cited by 18 (4 self)
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Abstract. We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.