A maximum entropy principle based closure method for macro-micro models of polymeric materials
Yunkyong Hyon José A. Carrillo Qiang Du Chun Liu
We consider the finite extensible nonlinear elasticity (FENE) dumbbell model in viscoelastic polymeric fluids. We employ the maximum entropy principle for FENE model to obtain the solution which maximizes the entropy of FENE model in stationary situations. Then we approximate the maximum entropy solution using the second order terms in microscopic configuration field to get an probability density function (PDF). The approximated PDF gives a solution to avoid the difficulties caused by the nonlinearity of FENE model. We perform the moment-closure approximation procedure with the PDF approximated from the maximum entropy solution, and compute the induced macroscopic stresses. We also show that the moment-closure system satisfies the energy dissipation law. Finally, we show some numerical simulations to verify the PDF and moment-closure system.
keywords: numerical simulations. non-newtonian fluid multiscale modeling FENE model maximum entropy principle polymeric fluid Fokker-Planck equation macro-micro dynamics moment closure
Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations
José A. Carrillo Jean Dolbeault Ivan Gentil Ansgar Jüngel
In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus $S^1$ and explore their implications for the long-time asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropy-entropy production methods and based on appropriate Poincaré type inequalities.
keywords: Poincare inequality higher-order nonlinear PDEs entropy production entropy-entropy production method Sobolev estimates entropy fast diffusion equation parabolic equations Logarithmic Sobolev inequality long-time behavior thin film equation. porous media equation
Explicit equilibrium solutions for the aggregation equation with power-law potentials
José A. Carrillo Yanghong Huang

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernelsare constructed by inverting Fredholm integral operators or byemploying certain integral identities. These solutions are expected tobe the global energy stable equilibria and to characterize the generic behaviorsof stationary solutions for more general interactions.

keywords: Fredholm integral equations stationary solutions
Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system
Blanca Ayuso José A. Carrillo Chi-Wang Shu
We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for all the proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.
keywords: discontinuous Galerkin energy conservation. mixed-finite elements Vlasov-Poisson system
Double milling in self-propelled swarms from kinetic theory
José A. Carrillo M. R. D’Orsogna V. Panferov
We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.
keywords: kinetic theory Interacting particle systems swarming milling patterns.
Over-populated tails for conservative-in-the-mean inelastic Maxwell models
José A. Carrillo Stéphane Cordier Giuseppe Toscani
We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of the authors [24], which are conservative in the mean but possess a steady distribution with Pareto tails. The convolution-like gain operator is subsequently shown to have good contraction/expansion properties with respect to different metrics in the set of probability measures. Existence and regularity of isotropic stationary states is shown directly by constructing converging iteration sequences as done in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high energy tails of the steady profile are derived from the basic property of contraction/expansion of metrics. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as $t\to\infty$ to the steady solution in these distances, which metricizes the weak convergence of measures. These results show that power-like tails in Maxwell models are obtained when the point-wise conservation of momentum and/or energy holds only globally.
keywords: inelastic collisions thick-tails. conservative in mean asymptotic behavior stochastic restitution coefficient
Confinement for repulsive-attractive kernels
Daniel Balagué José A. Carrillo Yao Yao
We investigate the confinement properties of solutions of the aggregation equation with repulsive-attractive potentials. We show that solutions remain compactly supported in a large fixed ball depending on the initial data and the potential. The arguments apply to the functional setting of probability measures with mildly singular repulsive-attractive potentials and to the functional setting of smooth solutions with a potential being the sum of the Newtonian repulsion at the origin and a smooth suitably growing at infinity attractive potential.
keywords: particle systems Aggregation equation confinement.
Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system
José A. Carrillo Renjun Duan Ayman Moussa
We are concerned with the global well-posedness of a two-phase flow system arising in the modelling of fluid-particle interactions. This system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the incompressible Euler equations for a dense phase (fluid) through the friction forcing. Global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state, and moreover an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the macro-micro decomposition and Kawashima's hyperbolic-parabolic dissipation argument. This result is generalized to the periodic case, when particles are in the torus, improving the rate of convergence to exponential.
keywords: rate of convergence. global well-posedness Two-phase flow system
Positive entropic schemes for a nonlinear fourth-order parabolic equation
José A. Carrillo Ansgar Jüngel Shaoqiang Tang
A finite-difference scheme with positivity-preserving and entropy-decreasing properties for a nonlinear fourth-order parabolic equation arising in quantum systems and interface fluctuations is derived. Initial-boundary value problems, the Cauchy problem and a rescaled equation are discussed. Based on this scheme we elucidate properties of the long-time asymptotics for this equation.
keywords: long-time behavior of discrete solutions. Finite difference method discrete entropy estimates discrete positive solutions
Fast-reaction limit for the inhomogeneous Aizenman-Bak model
José A. Carrillo Laurent Desvillettes Klemens Fellner
Solutions of the spatially inhomogeneous diffusive\linebreak Aizenmann-Bak model for clustering within a bounded domain with homogeneous Neumann boundary conditions are shown to stabilize, in the fast reaction limit, towards local equilibria determined by their monomer density. Moreover, the sequence of monomer densities converges to the solution of a nonlinear diffusion equation whose nonlinearity depends on the size-dependent diffusion coefficient. Initial data are assumed to be integrable, bounded and with a certain number of moments in size. The number density of clusters for the solutions is assumed to verify uniform bounds away from zero and infinity independently of the scale parameter.
keywords: nonlinear-diffusion equations entropy-based estimates fast-reaction limit duality arguments. coagulation-fragmentation equation

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