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Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

John W. Barrett, Endre Süli (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity...

Fluid-dynamic equations for reacting gas mixtures

Marzia Bisi, Maria Groppi, Giampiero Spiga (2005)

Applications of Mathematics

Starting from the Grad 13-moment equations for a bimolecular chemical reaction, Navier-Stokes-type equations are derived by asymptotic procedure in the limit of small mean paths. Two physical situations of slow and fast reactions, with their different hydrodynamic variables and conservation equations, are considered separately, yielding different limiting results.

Formal passage from kinetic theory to incompressible Navier–Stokes equations for a mixture of gases

Marzia Bisi, Laurent Desvillettes (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the...

Fractional Fokker-Planck-Kolmogorov type Equations and their Associated Stochastic Differential Equations

Hahn, Marjorie, Umarov, Sabir (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 35R11, 35R60, 35Q84, 60H10 Dedicated to 80-th anniversary of Professor Rudolf GorenfloThere is a well-known relationship between the Itô stochastic differential equations (SDEs) and the associated partial differential equations called Fokker-Planck equations, also called Kolmogorov equations. The Brownian motion plays the role of the basic driving process for SDEs. This paper provides fractional generalizations of the triple relationship between the driving process, corresponding...

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

Annales de l'I.H.P. Probabilités et statistiques

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process ( K ( t ) , i ( t ) , Y ( t ) ) on ( 𝕋 2 × { 1 , 2 } × ℝ 2 ) , where 𝕋 2 is the two-dimensional torus. Here ( K ( t ) , i ( t ) ) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y ( t ) is an additive functional of K , defined as ∫ 0 t v ( K ( s ) ) d s , where | v | ∼ 1 for small k . We prove that the rescaled process ( N ln N ) - 1 / 2 Y ( N t ) converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately...

Gelation in coagulation and fragmentation models.

Miguel Escobedo (2002)

RACSAM

We first present very elementary relations between climate and aerosols. The we introduce the homogeneous coagulation equation as a simple model to describe systems of merging particles like polymers or aerosols. We next give a recent result about gelation of solutions. We end with some related open questions.

General approximation method for the distribution of Markov processes conditioned not to be killed

Denis Villemonais (2014)

ESAIM: Probability and Statistics

We consider a strong Markov process with killing and prove an approximation method for the distribution of the process conditioned not to be killed when it is observed. The method is based on a Fleming−Viot type particle system with rebirths, whose particles evolve as independent copies of the original strong Markov process and jump onto each others instead of being killed. Our only assumption is that the number of rebirths of the Fleming−Viot type system doesn’t explode in finite time almost surely...

Generalized kinetic equations and effective thermodynamics

Pierre-Henri Chavanis (2004)

Banach Center Publications

We introduce a new class of nonlocal kinetic equations and nonlocal Fokker-Planck equations associated with an effective generalized thermodynamical formalism. These equations have a rich physical and mathematical structure that can describe phase transitions and blow-up phenomena. On general grounds, our formalism can have applications in different domains of physics, astrophysics, hydrodynamics and biology. We find an aesthetic connexion between topics (stars, vortices, bacteries,...) which were...

Generic principles of active transport

Mauro Mobilia, Tobias Reichenbach, Hauke Hinsch, Thomas Franosch, Erwin Frey (2008)

Banach Center Publications

Nonequilibrium collective motion is ubiquitous in nature and often results in a rich collection of intriguing phenomena, such as the formation of shocks or patterns, subdiffusive kinetics, traffic jams, and nonequilibrium phase transitions. These stochastic many-body features characterize transport processes in biology, soft condensed matter and, possibly, also in nanoscience. Inspired by these applications, a wide class of lattice-gas models has recently been considered. Building on the celebrated...

Gibbs–non-Gibbs properties for evolving Ising models on trees

Aernout C. D. van Enter, Victor N. Ermolaev, Giulio Iacobelli, Christof Külske (2012)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves differently from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each...

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