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Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates
The long-time asymptotics of certain nonlinear , nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4]
Asymptotic L1-decay of solutions of the porous medium equation to self-similarity
We consider the flow of gas in an N-dimensional porous medium with initial density v0NxO 0. The density vNx;tO then satisfies the nonlinear degenerate parabolic equa- tion vt E —v m where m> 1 is a
Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities
Abstract. We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2)
On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions
We investigate a Boltzmann equation for inelastic scattering in which the relative velocity in the collision frequency is approximated by the thermal speed. The inelasticity is given by a velocity
A well-posedness theory in measures for some kinetic models of collective motion
We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have
Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media
An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles
Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states
The results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter and several aspects of the NNLIF model are analysed: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case.
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, we
Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation
Abstract: In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation ut=−(uuxxx)x, modeling the evolution of the
Particle, kinetic, and hydrodynamic models of swarming
The role of the kinetic viewpoint is emphasized in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system.