1.
Direct Stimulation of Aggregation Phenomena - Maury
, Bertrand
We present here an algorithm to simulate the motion of rigid bodies subject to a
non-overlapping constraint, and which tend to aggregate when they get close to each other. The
motion is induced by external forces. Two types of forces are considered here: drift force induced
by the action of a surrounding fluid whose motion is prescribed, and stochastic forces modelling
random shocks of molecules on the surface of the bodies. The numerical approach fits into the
general framework of granular flow modelling.
2.
Geometric Compression Using Riemann Surface Structure - Villedieu
, P.; Simonin
, O.
The present paper is devoted to the kinetic modeling of coalescence in turbulent
gas-droplet flows. A new approach is proposed for the calculation of the collision probability, that
takes into account the correlations induced by the effect of the gas on the droplet motion. The key
ingredient is to replace the simple distribution function
f(1)p (t, x, v, r), which is classically used for
the description of a spray at the kinetic level, by the joint distribution function,
f(1)pg (t, x, v, u, r),
which explicitely depends on the fluctuating gas velocity u at the droplet position.
3.
Some Modelling Issues in the Theory of Fragmentation-Coagulation Systems - Collet
, Jean-François
This paper is meant as an introduction to some of the most classical models in the theory
of fragmentation-coagulation. The main models presented are the Becker-Döring, fragmentation-coagulation
(discrete or continuous) and Lifshitz-Slyozov ones. Rather than focusing on mathematical
technicalities, we have chosen to insist on the physical ideas behind their derivation, in order
to present them in a unified framework. The unifying physical principle in this context is the mass
action principle, which we expose in detail, our philosophy being that these models may be thought
of as technical variations on this theme. We then present some qualitative properties of the models,
which include saturation, criticality, and...
4.
On a Discrete Boltzmann-Smoluchowki Equation with Rates Bounded in the Velocity Variables - Fourneir
, Nicholas; Mischler
, Stephane
Consider a spatially homogeneous infinite particle system in which coalescence and
elastic collisions occur. The Boltzmann-Smoluchowski equation describes the evolution of the concentration
f(t, m, v) of particles of mass m and velocity v at time t ? 0. Using a stochastic version
of this equation, we give an exact simulation scheme and we study the asymptotics of solutions for
large times.
5.
Apearance of Dust in Fragmentations - Haas
, Benedicte
For fragmentations in which particles split even faster when their mass is smaller, it
is possible to observe a decrease of the total mass of the system, due to the reduction into dust. We
investigate here this appearance of dust for a large class of deterministic and random fragmentations.
6.
Uniqueness via Probabilistic Interpretation for the Discrete Coagulation Fragmentation Equation - Jourdain
, Benjamin
In this paper, supposing that either the initial data is small or the fragmentation
phenomenon dominates the coagulation, we associate a nonlinear stochastic process with any solution
of the mass-flow equation obtained from the discrete Smoluchowski coagulation fragmentation
equation by a natural change of variables. This enables us to deduce uniqueness for the mass flow
equation and therefore for the corresponding Smoluchowski equation thanks to a coupling argument.
7.
Macroscopic Limits of the Becker-Döring Equations - Niethammer
, Barbara
We review the derivation of macroscopic limits of the Becker-Döring equations. We
show that those limits have the structure of a gradient flow even though the Becker-Döring equations
themselves do not allow for such an interpretation.
8.
Brownian Coagulation - Norris
, J.R.
We consider a stochastic particle model for coagulating particles, whose free motion
is Brownian, with diffusivity given by Einstein's law. We present in outline a derivation from this
model of a spatially inhomogeneous version of Smoluchowski's coagulation equation. Some analytic
results on existence, uniqueness and mass conservation for the limit equation are also presented.
9.
Global Weak Solutions to the Relativistic
Vlasov-Maxwell System Revisited - Rein
, Gerhard
In their seminal work [3], R. DiPerna and P.-L. Lions established the existence of global weak solutions
to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result
for the relativistic version of this system, the main purpose being to make this important result
of kinetic theory more easily accessible to newcomers in the field.
We show that the weak solutions preserve the total charge.
10.
Dynamic Bifurcation and Stability in the
Rayleigh-Benard Convection - Ma
, Tian; Wang
, Shouhong
We study in this article the bifurcation and stability of the solutions
of the Boussinesq equations, and the onset of the Rayleigh-Benard convection.
A nonlinear theory for this problem is established in this article using a new notion of bifurcation
called attractor bifurcation and its corresponding theorem developed recently by the authors in [6]. This
theory includes the following three aspects. First, the problem bifurcates from the trivial solution
an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number
Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue
Rc for the linear problem.
Second, the bifurcated attractor AR is...
11.
Removing the Cell Resonance Error in the
Multiscale Finite Element Method via a Petrov-Galerkin Formulation - Hou, Thomas Y.; Wu, Xiao-Hui; Zhang, Yu
We continue the study of the nonconforming multiscale finite element method
(Ms-FEM) introduced in [17, 14] for second order elliptic equations with
highly oscillatory coefficients.
The main difficulty in MsFEM, as well as other numerical upscaling methods,
is the scale resonance effect. It has been show that the leading order resonance error can be
effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell
resonance error of O(e2h2).
Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions
and linear test functions. We show that the cell resonance error is eliminated in this formulation
and hence the convergence rate is greatly improved. Moreover,...
12.
Nearly Lipschitzean Divergence Free Transport Propogates neither Continuity nor BV Regularity - Colombini, Ferruccio; Luo, Tao; Rauch, Jeffrey
We give examples of divergence free vector fields.
For such fields the Cauchy problem for the linear transport equation has unique bounded solutions.
The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics.
In addition, there are smooth initial data so that the unique bounded solution is not continuous on any neighborhood of the origin.
The second example is a field of similar regularity and intial data of bounded variation.
13.
Coninuous Glimm-Type Functionals and Spreading of Rarefaction Waves - Lefloch, Philippe G.; Trivisa, Andkonstantina
Several Glimm-type functionals for (piecewise smooth) approximate solutions
of nonlinear hyperbolic systems have been introduced in recent years.
In this paper, following a work by Baiti and Bressan on genuinely nonlinear systems
we provide a framework to prove that such functionals can be extended to general functions
with bounded variation and we investigate their lower semi-continuity properties
with respect to the strong L1topology. In particular, our result applies
to the functionals introduced by Iguchi-LeFloch and Liu-Yang for systems
with general flux-functions, as well as the functional introduced
by Baiti-LeFloch-Piccoli for nonclassical entropy solutions.
As an illustration of the use of continuous Glimm-type functionals,
we also extend a result by Bressan...
14.
G-Norm Properties of Bounded Variation Regularization - Osher, Stanley; Scherzer, Otmar
Recently Y. Meyer derived a characterization of the minimizer of
the Rudin-Osher-Fatemi functional in a functional analytical framework.
In statistics the discrete version of this functional is used
to analyze one dimensional data and belongs to the class of nonparametric regression models.
In this work we generalize the functional analytical results of Meyer and apply them
to a class of regression models, such as quantile, robust, logistic regression,
for the analysis of multidimensional data. The characterization of Y. Meyer
and our generalization is based on G-norm properties of the data and the minimizer.
A geometric point of view of regression minimization is provided.
15.
Multiscale Couplings In Prototype Hybrid
Deterministic/Stochastic Systems:
Part I, Deterministic Closures - Katsoulakis, M. A.; Majda, A. J.; Sopasakis, A.
We introduce and study a class of model prototype hybrid systems
comprised of a microscopic stochastic surface process modeling adsorption/desorption
and/or surface di.usion of particles coupled to an ordinary di.erential equation (ODE)
displaying bifurcations excited by a critical noise parameter.
The models proposed here are caricatures of realistic systems arising in diverse applications
ranging from surface processes and catalysis to atmospheric and oceanic models.
We obtain deterministic mesoscopic models from the hybrid system by employing two methods:
stochastic averaging principle and mean field closures. In this paper we focus on the case
where phase transitions do not occur in the stochastic system.
In the averaging principle case a faster...
16.
Analysis of 1 + 1 Dimensional Stochastic Models of Liquid Crystal Polymer Flows - Li, Tiejun; Zhang, Pingwen; Zhou, Xiang
We consider the stochastic model of concentrated Liquid Crystal Polymers(LCPs) in the plane Couette flow.
The dynamic equation for the liquid crystal polymers is described
by a nonlinear stochastic differential equation with Maier-Saupe interaction potential.
The stress tensor is obtained from an ensemble average of microscopic polymer configurations.
We present the local existence and uniqueness theorem for the solution of the coupled fluid-polymer system.
We also analyze the error of a fully .nite di.erence-Monte Carlo hybrid numerical scheme
by investigating the asymptotic behavior of weakly interacting processes.
The rate of convergence of the full discretized scheme is derived.
17.
Burgers' Equation with Vanishing Hyper-Viscosity - Tadmor, Eitan
We prove that bounded solutions of the vanishing hyper-viscosity equation,
converge to the entropy solution of the corresponding convex conservation law.
The hyper-viscosity case lacks the monotonicity which underlines
the Krushkov BV theory in the viscous case s = 1.
Instead we show how to adapt the Tartar-Murat compensated compactness theory together
with a weaker entropy dissipation bound to conclude the convergence of the vanishing hyper-viscosity.
18.
A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems - Kurtz, Thomas G.; Xiong, Jie
In an earlier paper, we studied the approximation of solutions
V(t) to a class of SPDEs by the empirical measure Vn(t)
of a system of n interacting diffusions.
In the present paper, we consider a central limit type problem,
showing that \sqrt{n}(Vn-V )n converges weakly,
in the dual of a nuclear space, to the unique solution of a stochastic evolution equation.
Analogous results in which the di.usions that determine Vn
are replaced by their Euler approximations are also discussed.
19.
Gravity driven shallow water models for arbitrary topography - Bouchut, Francois; Westdickenberg, Michael
We derive new models for gravity driven shallow water flows
in several space dimensions over a general topography.
A first model is valid for small slope variation, i.e. small curvature,
and a second model is valid for arbitrary topography.
In both cases no particular assumption is made on the velocity profile in the material layer.
The models are written for an arbitrary coordinate system,
and several formulations are provided.
A Coulomb friction term is derived within the same framework,
relevant in particular for debris avalanches. All our models are invariant under rotation,
admit a conservative energy equation, and preserve the steady state of a lake at rest.
20.
Domain decomposition algorithm for the parabolic equation with variable coefficient - Sheng, Zhiqiang; Liu, Xingping; Cui, Xia
In this paper, we design a domain decomposition algorithm for the two-dimensional parabolic equation
with variable coefficient by using a larger spacing at interface points and
the implicit scheme at the interior points, hence get an algorithm with the relaxed stability bounds.
Then we prove the stability and analyze the accuracy of the algorithm by using the idea of maximum principle.
Some results of numerical experiments are also provided.
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