FluidKinetic Modelling in Biology, Physics & Engineering
Monday 6th September 2010 to Friday 10th September 2010
08:30 to 09:55  Registration  
09:55 to 10:00  Welcome from Sir David Wallace (INI Director)  
10:00 to 11:00 
F Golse (École Polytechnique) Homogenization of the linear Boltzmann equation in a periodic system of holes
Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed &. Assuming that the holes are absorbing  i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of freepath lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system decays exponentially in the long time limit. This is at variance with the collisionless case discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), 199221], where the total mass decays as C/t in the long time limit. (Work in collaboration with E. Bernard and E. Caglioti.)

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Selfsimilarity in coagulation equations with nonlocal drift
In this talk we consider kinetic equations that model coarsening phenomena which involve transport of mass and rearrangement due to coalescence. One expects that solutions converge in the largetime regime to selfsimilar form. However, due to the nonlocal terms in the equations, the study of selfsimilar solutions is not straightforward. We discuss several strategies that allow to establish existence, uniqueness and decay properties of selfsimilar solutions.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:45 to 14:45 
Y Brenier ([Nice]) A modified least action principle allowing mass concentrations for the early universe reconstruction problem
We discuss the reconstruction problem for the early universe, following Peebles and Frisch and coauthors. The model is given by the pressureless gravitational Euler Poisson system, with time dependent coefficients taking into account general relativity features. (This amounts to considering Newtonian gravation in an Einstein de Sitter background.) The standard least action principle is unable to take into account mass concentration effects. We show that a modification of the action is possible which allows dynamical concentrations. (detailed discussion available on HAL preprint server.)

INI 1  
14:45 to 15:25 
Orbital stability of spherical galactic models
We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov in 1961. In a previous work (arXiv:0904.2443), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature. In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.
This is a joint work with Mohammed Lemou and Pierre Raphael.

INI 1  
15:25 to 15:45  Afternoon Tea  
15:45 to 16:25 
Scalings for a ballistic aggregation equation
We consider a mean field type equation for ballistic aggregation of particles whose density function depends both on the mass and impulsion of the particles. For the case of a constant aggregation rate the existence of selfsimilar solutions and the convergence of more general solutions to them is proven.The large time decay of some moments of general solutions is estimated. Some new classes of selfsimilar solutions for several classes of mass and/or impulsion dependent rates are obtained.

INI 1  
16:25 to 17:05 
M Burger ([Münster]) Nonlinear CrossDiffusion Models for Size Exclusion
In this talk we discuss nonlinear crossdiffusion models arising from size exclusion effects in biological processes with heterogeneous species. We discuss the macoscopic modeling and basic issues in the analysis of the arising systems of partial differential equations, which is a challenging task due to degeneracy and absence of maximum principles. Moreover we present applications to ion transport in channels, cell migration, and swarm models.

INI 1  
17:05 to 17:45 
Analysis of Dynamics of DoiOnsager Phase Transition
Phase transition of directional field appears in some physical and biological systems such as ferromagnetism near Currie temperature, flocking dynamics near critical mass of self propelled particles. This problem was postulated by Onsager via minimization of a free energy and dynamically by Doi equation. It also appears in the meanfield equation of Vicsek model for flocking of birds. In this talk, I will present a new entropy for the DoiOnsager equation which enable us to give a rigorous justification of this dynamics phase transition. This is a joint work with Pierre Degond and Amic Frouvelle.

INI 1  
17:45 to 18:30  Welcome Wine Reception  
18:45 to 19:30  Dinner at Wolfson Court 
09:00 to 10:00 
K Aoki ([Kyoto]) Some decay problems of a collisionless gas: Numerical study
We investigate timedependent behavior of a collisionless (or highly rarefied) gas in the following two problems:
(i) A collisionless gas is confined in a closed domain bounded by a diffusely reflecting wall with a uniform temperature. The approach of the gas to an equilibrium state at rest, caused by the interaction of gas molecules with the wall, is investigated numerically. It is shown that the approach is slow and proportional to an inverse power of time. This is a joint work with T. Tsuji and F. Golse.
(ii) An infinite plate without thickness is placed in a collisionless gas, and an external force, obeying Hooke's law, is acting perpendicularly on the plate. If the plate is displaced perpendicularly from its equilibrium position and released, then it starts an oscillatory motion, which decays as time goes on because of the drag exerted by the gas molecules. This unsteady motion is investigated numerically, under the diffuse reflection condition, with special interest in the manner of its decay. It is shown that the decay of the displacement of the plate is slow and is in proportion to an inverse power of time. The result complements the existing mathematical study of a similar problem [S. Caprino, et al., Math. Models. Meth. Appl. Sci. 17, 1369 (2007)] in the case of nonoscillatory decay. This is a joint work with T. Tsuji.

INI 1  
10:00 to 11:00 
L Desvillettes (ENS de Cachan) Some new results of existence for the theory of sprays
We present results of existence for sprays equations. Those are systems in which a Vlasov (or VlasovFokkerPlanck, or VlasovBoltzmann) equation is coupled with an equation of fluid mechanics (compressible or incompressible Euler; compressible or incompressible Navierstokes; or even Boltzmann). They model complex flows in which a dispersed (liquid or solid) phase interacts with a gas. We detail the latest results, obtained with Laurent Boudin, Céline Granmont and Ayman Moussa, and corresponding to incompressible viscous thin sprays.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Relative entropy method applied to the stability of shocks for systems of conservation laws
We develop a theory based on relative entropy to show stability and uniqueness of extremal entropic RankineHugoniot discontinuities for systems of conservation laws (typically 1shocks, nshocks, 1contact discontinuities and ncontact discontinuities of big amplitude), among bounded entropic weak solutions having an additional strong trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, the strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Viscosity of bacterial suspensions: experiment and theory
Measurements of the shear viscosity in suspensions of swimming Bacillus subtilis in freestanding liquid films have revealed that the viscosity can decrease by up to a factor of 7 compared to the viscosity of the same liquid without bacteria or with nonmotile bacteria. The viscosity depends on the concentration and swimming speed of the bacteria.
The effective viscosity of dilute suspensions of swimming bacteria from the microscopic details of the interaction of an elongated body with the background flow is derived. An individual bacterium propels itself forward by rotating its flagella and reorients itself randomly by tumbling. Due to the bacterium’s asymmetric shape, interactions with a background flow cause the bacteria to preferentially align in directions in which selfpropulsion produces a significant reduction in the effective viscosity.
1. Andrey Sokolov and Igor S. Aranson, Reduction of Viscosity in Suspension of Swimming Bacteria, Phys. Rev. Lett. 103, 148101 (2009) 2. Brian M. Haines, Andrey Sokolov, Igor S. Aranson, Leonid Berlyand, and Dmitry A. Karpeev, Threedimensional model for the effective viscosity of bacterial suspensions, Phys. Rev. E 80, 041922 (2009)

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:10 
A E Tzavaras ([Maryland]) Kinetic models in material science
I will review some recent works on the derivation and study of kinetic models in a
context of material science problems:
(i) the derivation of kinetic equations from a class of particle systems that describes theories for
crystalline interfaces. In this line of work we derive the macroscopic limits of theories that describe
crystal interfaces starting from models at the nanoscale from the perspective of kinetic theory.
(joint work with Dio Margetis, Univ. of Maryland)
(ii) the study of certain kinetic equations that appear in modeling sedimentation for dilute suspensions
for rigid rods. Here, we study a class of models introduced by Doi and describing suspensions
of rod{like molecules in a solvent
uid. Such models couple a microscopic FokkerPlanck type
equation for the probability distribution of rod orientations to a macroscopic Stokes
ow. We
show that steady states can have discontinuous solutions analogous to the ones studied in the
context for macroscopic viscoelastic models (e.g. for OldroydB models) and spurt phenomena
or shear bands in that context. Also, that the longtime behavior of the sedimentating
ow is
approximated in a diusive scaling by the KellerSegel model. (joint work with Ch. Helzel, U.
Bochum and F. Otto, Leipzig).

INI 1  
16:10 to 16:50 
Some polymeric fluid flow models: steady states and largetime convergence
We consider a dumbbell model for a dilute solution of polymers in a homogeneous fluid. In a micromacro model, the incompressible NavierStokes equation for the fluid flow is coupled to a FokkerPlanck equation for the (microscipic) distribution of the polymeric chains.
First we analyze the linear FokkerPlanck equation for Hookean dumbbells and in the case of finite extension nonlinear elasticity (FENE): steady states and largetime convergence using entropy methods. In the FENE case the stationary problem is degenerate elliptic, requiring to use weighted Sobolev spaces. In the coupled Hookean case we also show exponential convergence to a homogeneous stationary flow.
References: A. Arnold, J.A. Carrillo, C. Manzini: Refined longtime asymptotics for some polymeric fluid flow models, Comm. Math. Sc. 8, No. 3 (2010) 763782.
A. Arnold, C. Bardos, I. Catto: Stable steady states of a FENEdumbbell model for polymeric fluid flows, preprint, 2010.

INI 1  
16:50 to 17:30 
LBM: Approximate Invariant Manifolds and Stability
We study the Lattice Boltzmann Models in the framework of the Geometric Singular Perturbation theory. We begin with the Lattice Boltzmann system discrete in both velocity space and time with the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes. When time step is small then this system has an approximate invariant manifold close to locally equilibrium distributions. We found a time step expansion for the approximate invariant manifold and proved its conditional stability in any order of accuracy under condition that the space derivatives of the correspondent order remain bounded. On this invariant manifold, a macroscopic dynamics arises and we found the time step expansion of the equation of the macroscopic dynamics.

INI 1  
17:30 to 18:00 
J Haskovec (Austrian Academy of Sciences) A stochastic individual velocity jump process modelling the collective motion of locusts
We consider a model describing an experimental setting, in which locusts run in a ringshaped arena. With intermediate spatial density of the individuals, coherent motion is observed, interrupted by sudden changes of direction ("switching"). Contrary to the known model of Czirok and Vicsek, our model assumes runs of the individuals in either positive or negative direction of the 1D arena with the same speed, that are subject to random switches. As supported by experimental evidence, the individual switching frequency increases in response to a local or global loss of group alignment, which constitutes a mechanism to increase the coherence of the group. We show that our individual based model, although phenomenologically very simple, exhibits nontrivial dynamics with a "phase change" behaviour, and, in particular, recovers the observed group directional switching. Passing to the corresponding FokkerPlanck equation, we are able to give estimates of the expected switching times in terms of number of individuals and values of the model coefficients. Then we pass to the kinetic description, recovering a system of two kinetic equations with nonlocal and nonlinear right hand sides, which is valid when the number of individuals tends to infinity. We perform a mathematical analysis of the system, show some numerical results and point out several interesting open problems.

INI 1  
18:00 to 18:30 
Asymptotic dynamics of a population density: a model with a survival threshold
We study the long timelong range behavior of reaction diffusion equations with negative squareroot reaction terms. In particular we investigate the exponential behavior of the solutions after a standard hyperbolic scaling. This leads to a HamiltonJacobi equality with an obstacle that depends on the solution itself. Our motivation comes from the socalled “tail problem” in population biology. We impose extramortality below a given survival threshold to avoid meaningless exponential tails. This is a joint work with G. Barles, B. Perthame and P. E. Souganidis.

INI 1  
18:45 to 19:30  Dinner at Wolfson Court 
09:00 to 10:00  Hydrodynamic limits, Knudsen layers and Numerical fluxes  INI 1  
10:00 to 11:00 
Macroscopic limits and decay to equilibrium for kinetic equations with relaxation collision kernels and mass conservation
Recent results on the rigorous derivation of nonlinear convection diffusion equations (ranging from porous medium to fast diffusion models) from kinetic transport models will be reviewed. Entropy dissipation techniques are employed in combination with compensated compactness.
Exponential convergence to global equilibria is derived by a general abstract strategy for proving hypocoercivity results. It relies on the construction of a Lyapunov functional by a modification of the quadratic relative entropy and is inspired by the theory of hypoellipticity.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Novel phenomena and models of active fluids
Fluids with suspended microstructure  complex fluids  are common actors in micro and biofluidics applications and can have fascinating dynamical behaviors. A new area of complex fluid dynamics concerns "active fluids"
which are internally driven by having dynamic microstructure such as swimming bacteria. Such motile suspensions are important to biology, and are candidate systems for tasks such as microfluidic mixing and pumping.
To understand these systems, we have developed both firstprinciples particle and continuum kinetic models for studying the collective dynamics of hydrodynamically interacting microswimmers. The kinetic model couples together the dynamics of a Stokesian fluid with that of an evolving "active" stress field. It has a very interesting analytical and dynamical structure, and predicts critical conditions for the emergence of hydrodynamic instabilities and fluid mixing. These predictions are verified in our detailed particle simulations, and are consistent with current experimental observation.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
P Maini ([Oxford]) Modelling Aspects of Solid Tumour Growth
We present a number of models proposed to address different aspects of solid tumour growth. A simple partial differential equation model is shown to make experimentally verified predictions on how certain tumours invade due to the acidic environments they create through metabolism. A hybrid cellular automaton model is presented to investigate somatic evolution and, finally, a multiscale model for tumour vasculogenesis is presented, encorporating blood vessel structural adaption and angiogenesis.

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:10 
N Bournaveas ([Edinburgh]) Kinetic models of chemotaxis
Chemotaxis is the directed motion of cells towards higher concentrations of chemoattractants. At the microscopic level it is modeled by a nonlinear kinetic transport equation with a quadratic nonlinearity. We'll discuss global existence results obtained using dispersion and Strichartz estimates, as well as some blow up results. (joint work with Vincent Calvez, Susana Gutierrez and Benoit Perthame).

INI 1  
16:10 to 16:50 
V Calvez (École Normale Supérieure) Kinetic models for bacterial chemotaxis
In this talk we discuss kinetic models for selforganization of cells. The framework is the OthmerDunbarAlt model. It describes a velocityjump process accounting for the motion of cells (e.g. bacteria) biased by a chemical signal. Several behaviours can be observed, ranging from blowup in finite time to propagation of traveling waves. Blowup is proved for a specific choice of the model under radial symmetry. The issue of traveling bands of bacteria is addressed through a more involved model. Comparison to experimental data is also discussed.
This is a joint work with N. Bournaveas (Univ. Edinburgh), B. Perthame (Univ. Paris 6 "Pierre & Marie Curie"), and A. Buguin, J. Saragosti, P. Silberzan (Institut Curie, Paris).

INI 1  
16:50 to 17:30 
Cross diffusion preventing blow up in the twodimensional KellerSegel model
We analyse the parabolic KellerSegel system with an additional crossdiffusion term guaranteeing globalintime existence of weak solutions for large data. This modification provides another helpful entropy dissipation term, which is used to show the global existence of solutions for any initial mass. For the proof we first analyse an approximate problem obtained from a semidiscretisation and a carefully chosen regularisation by adding higher order derivatives. Compactness arguments are used to carry out the limit to the original system. Our model also allows for further entropy estimates and may be helpful in numerical simulations to detect the occurence of blowup.

INI 1  
17:30 to 18:00 
An integrodifferential model to study evolution
The evolution of a population structured by a continuous phenotypic trait can be modeled by integrodifferential models called "Kimura models". Using numerical simulations, this model shows that the population often concentrates around a finite number of traits, which correspond to different species. In this presentation, we will present long time and rare mutations asymptotic results for this type of models.

INI 1  
19:30 to 22:00  Conference Dinner at Emmanuel College 
09:00 to 10:00 
Kinetic description and connectivity of old and new models of flocking
We introduce a new particlebased model for flocking and we show that, as with the CuckerSmale model, flocking occurs when pairwise long range interactions are sufficiently strong. Next, we derive a Vlasovtype kinetic model for these particle models and we inquire about their timeasymptotic flocking behavior for compactly supported initial data and the connectivity of their underlying graph. Finally, we introduce a hydrodynamic descriptions of flocking based on the kinetic models and show flocking behavior without closure of higher moments.

INI 1  
10:00 to 11:00 
Mathematical Modeling of Selection and Orientational Aggregation in Population Dynamics
During this talk equations of transport type for interacting cell populations will be discussed, where an additional parameter for the respective cell stage is taken into account. We are interested in the qualitative behavior of these models in the context of cell differentiation and aggregation, and will analyze under which conditions the system selects a finite number of cell stages from the continuum as long time behavior.
The talk summarizes joint works with K. Kang, B. Perthame, I. Primi, and J.J.L. Velazquez.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
P Souganidis ([Chicago]) Scalar conservation laws with rough (stochastic) fluxes and stochastic averaging lemmas
I will present a new theory of stochastic entropy solutions for scalar conservation laws with rough path (stochastic) fluxes. I will also discuss stochastic averaging lemmas that lead to new regularizing effects. This is joint work with B. Perthame and P. L. Lions.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:10 
S Fedotov ([Manchester]) Subdiffusion and nonlinear reactiontransport equations
The main aim of the talk is to discuss how to incorporate the nonlinear kinetic term into nonMarkovian transport equations described by a continuous time random walk (CTRW).
We derive nonlinear Master equations for the mean density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions.
We apply these equations to the problem of front propagation in the reactiontransport systems.
Applications include the transport of particles in spiny dendrites, the proliferation and migration dichotomy of the tumor cell invasion.

INI 1  
14:10 to 14:50 
Wave propagation in nonhomogeneous semilinear reactiondiffusion
The selection of the speed of a wavefront through which a stable state invades an unstable one has been very widely studied (since Fisher and KPP) for homogeneous scalar semilinear reactiondiffusion equations. Respects in which the homogeneous case is nongeneric when embedded within a heterogeneous setting will be highlighted.

INI 1  
14:50 to 15:30  Afternoon Tea  
15:30 to 16:00 
Coupled ChemotaxisFluid Models
We consider coupled chemotaxisfluid models aimed to describe swimming bacteria, which show bioconvective flow patterns on length scales much larger than the bacteria size. This behaviour can be modelled by a system consisting of chemotaxis equations coupled with viscous incompressible fluid equations through transport and external forcing. The globalintime existence of solutions to the Cauchy problem in two and three space dimensions is established. Precisely, for the chemotaxisNavierStokes system, we obtain global existence and convergence rates of classical solutions near constant states. When the fluid motion is described by Stokes equations, we derive some free energy functionals to prove globalintime existence of weak solutions for cell density with finite mass, firstorder spatial moment and entropy provided that the potential is weak or the substrate concentration is small. Moreover, with nonlinear diffusion for the bacteria, we give globalintime existence of weak solutions in two space dimensions.

INI 1  
16:00 to 16:30 
LM Tine ([INRIA Lille]) Existence and uniqness of solution to the LifshitzSlyozov system with monomers spatial diffusion
The standard LifshitzSlyozov model describes the evolution of a population of macroparticles or polymers immersed in a bath of monomers. It appears in such solution interaction phenomena between macroparticles characterized by their size density repartition f (t, ξ ) and the monomers characterized by their concentration c(t). These interactions induce the growth of large particles at the expense of the smaller ones what is known as Ostwald ripening. The evolution dynamic is governed by partial diﬀerential equations.
We extend this standard model to a more complexe one taking into account the spatial diﬀusion of the monomers concentration. So we prove the existence and uniqueness of solution for the model.

INI 1  
16:30 to 17:30 
FokkerPlanck models for BoseEinstein particles
We study nonnegative, measurevalued solutions of the initial value problem for onedimensional driftdiffusion equations where the linear drift has a driving potential with a quadratic growth at infinity, and the nonlinear diffusion is governed by an increasing continuous and bounded function. The initial value problem is studied in correspondence to initial densities that belong to the space of nonnegative Borel measures with finite mass and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the largetime behavior of solutions depends on a critical mass which can be explicitly characterized in terms of the diffusion function and of the drift term. If the initial mass is less than the critical mass, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass is accumulated.

INI 1  
17:30 to 18:00  Wine Reception  
18:45 to 19:30  Dinner at Wolfson Court 
09:00 to 10:00 
Fluid models of swarming behavior
Swarming biological systems (such as fish schools, insect swarms, mammalian herds) exhibit large scale spatiotemporal coordinated structures such as congestions, waves, oscillations, etc. The selforganization behavior is not directed encoded in the local interactions between individuals and emerges when the number of agents is large. In this situation, it is legitimate to use fluid models. However, the relation between the microscopic agent level and the macroscopic fluid level is not as straightforward as in the classical gas dynamics case. In this talk, we will review a certain number of the mathematical problems posed by these systems and some of the answers that can be given.

INI 1  
10:00 to 11:00 
On the time discretization of kinetic equations in stiff regimes
We review some results concerning the time discretization of kinetic equations in stiff regimes and their stability properties. Such properties are particularly important in applications involving several lenght scales like in the numerical treatment of fluidkinetic regions. We emphasize limitations presented by several standard schemes and focus our attention on a class of exponential RungeKutta integration methods. Such methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation operators of BGK type. For Boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
A class of selfsimilar solutions for the VlasovEinstein system (joint work with A. Rendall)
In this talk I will describe a family of spherically symmetric selfsimilar solutions for the massless EinsteinVlasov system. These solutions are supported in some particular hypersurfaces of the phase space and therefore, they are measurelike solutions, not fully dispersive.
The construction of these solutions relies in reformulating the problem as a fourdimensional dynamical systems problem that is studied in detail in a suitable perturbative limit.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 14:40 
Trend to the equilibrium for kinetic FokkerPlanck equations on Riemannian manifolds
In this talk I will present a result concerning the rate of convergence to the equilibrium for a class of degenerate transportdiffusive equations with periodic boundary conditions in the spatial variable. The diffusive part is given by the LaplaceBeltrami operator associated to a positive definite metric. Under suitable conditions on the velocity field and the Ricci curvature of the metric, all solutions convergence exponentially fast in time to the unique equilibrium state. The proof is by estimating the time derivative of the "modified" entropy in the formalism of Riemannian geometry.

INI 1  
14:40 to 15:10  Afternoon Tea  
15:10 to 15:50 
DSMCfluid solver with dynamic localisation of kinetic effects
In the present work, we present a novel numerical algorithm to couple the Direct Simulation Monte Carlo method (DSMC) for the solution of the Boltzmann equation with a finite volume like method for the solution of the Euler equations.
Recently we presented in [1],[2],[3] different methodologies which permit to solve fluid dynamics problems with localized regions of departure from thermodynamical equilibrium. The methods rely on the introduction of buffer zones which realize a smooth transition between the kinetic and the fluid regions.
In this talk we extend the idea of buffer zones and dynamic coupling to the case of the Monte Carlo methods. To facilitate the coupling and avoid the onset of spurious oscillations in the fluid regions which are consequence of the coupling with a stochastic numerical scheme, we use a new technique which permits to reduce the variance of the particle methods [4]. In addition, the use of this method permits to obtain estimations of the breakdowns of the fluid models less affected by fluctuations and consequently to reduce the kinetic regions and optimize the coupling.
[1] P.Degond, S.Jin, L. Mieussens, A Smooth Transition Between Kinetic and Hydrodynamic Equations , Journal of Computational Physics, Vol. 209, pp. 665694, (2005).
[2] P.Degond, G. Dimarco, L. Mieussens., A Moving Interface Method for Dynamic Kineticfluid Coupling. J. Comp. Phys., Vol. 227, pp. 11761208, (2007).
[3] P.Degond, G. Dimarco, L. Mieussens., A Multiscale KineticFluid Solver with Dynamic Localization of Kinetic Effects. J. Comp. Phys., Vol. 229, pp.49074933, (2010).
[4] P. Degond, G. Dimarco, L. Pareschi, The Moment Guided Monte Carlo Method, To appear in International Journal for Numerical Methods in Fluids.

INI 1  
15:50 to 16:30 
Aggregationpattern due to repulsiveaggregating interaction potentials
We study nonlocal evolution equations for a density of individuals, which interact through a given symmetric potential. Such models appear in many applications such as swarming and flocking, opinion formation, inelastic materials, .... In particular, we are interested in interaction potentials, which behave locally repulsive, but aggregating over large scales. A particular example for such potentials was recently given in models of the alignment of the directions of filaments in the cytoskeleton.
We present results on the structure and stability of steady states. We shall show that stable stationary states of regular interaction potentials generically consist of sums of Dirac masses. However the amount of Diracs depends on the interplay between local repulsion and aggregation. In particular we shall see that singular repulsive interaction potentials introduce diffusive effects in the sense that stationary state are no longer sums of Diracs but continuous densities. This is comparted to effects of added linear diffusion.

INI 1  
16:30 to 17:00 
P Carcaud ([Rennes 1]) Study of two simplified kinetic models for evaporation in gravitation 
INI 1  
17:00 to 17:30 
Large time behavior of collisionless plasma
The motion of a collisionless plasma, a hightemperature, lowdensity, ionized gas, is described by the VlasovMaxwell equations. In the presence of large velocities, relativistic corrections are meaningful, and when magnetic effects are neglected this formally becomes the relativistic VlasovPoisson system. Similarly, if one takes the classical limit as the speed of light tends to inﬁnity, one obtains the classical VlasovPoisson system. We study the longtime dynamics of these systems of PDE and contrast the behavior when considering the cases of classical versus relativistic velocities and the monocharged (i.e., single species of ion) versus neutral plasma situations.

INI 1  
18:45 to 19:30  Dinner at Wolfson Court 