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Seminars (KIT)

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Event When Speaker Title Presentation Material
KITW04 25th August 2010
09:00 to 10:30
Fokker-Planck models for Bose-Einstein particles
KITW04 25th August 2010
11:00 to 12:30
Regularity theory for non linear singular integral operators based on De Giorgi techniques
KITW04 26th August 2010
09:00 to 10:30
Regularity theory for non linear singular integral operators based on De Giorgi techniques II
KITW04 26th August 2010
11:00 to 12:30
S Fedotov Non-Markovian walks and nonlinear reaction-transport equations
KITW04 27th August 2010
09:00 to 10:30
Kinetic models in wealth distribution II
KITW04 27th August 2010
11:00 to 12:30
Front propagation in reaction-dispersal systems
KITW04 31st August 2010
09:00 to 10:30
Asymptotic-preserving schemes for some kinetic equations
KITW04 31st August 2010
11:00 to 12:30
Semi-classical models in semiconductor physics
KITW04 3rd September 2010
09:00 to 10:30
Semi-classical models in semiconductor physics II
KITW04 3rd September 2010
11:00 to 12:30
Estimate on commutator of the Laplacian and Leray projection operators and application to Navier-Stokes equation in bounded domain
KITW01 6th September 2010
10:00 to 11:00
F Golse 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 free-path 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), 199--221], where the total mass decays as C/t in the long time limit. (Work in collaboration with E. Bernard and E. Caglioti.)
KITW01 6th September 2010
11:30 to 12:30
Self-similarity 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 large-time regime to self-similar form. However, due to the nonlocal terms in the equations, the study of self-similar solutions is not straightforward. We discuss several strategies that allow to establish existence, uniqueness and decay properties of self-similar solutions.
KITW01 6th September 2010
13:45 to 14:45
Y Brenier 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 pressure-less 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 pre-print server.)
KITW01 6th September 2010
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.
KITW01 6th September 2010
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 self-similar 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 self-similar solutions for several classes of mass and/or impulsion dependent rates are obtained.
KITW01 6th September 2010
16:25 to 17:05
M Burger Nonlinear Cross-Diffusion Models for Size Exclusion
In this talk we discuss nonlinear cross-diffusion 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.
KITW01 6th September 2010
17:05 to 17:45
Analysis of Dynamics of Doi-Onsager 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 mean-field equation of Vicsek model for flocking of birds. In this talk, I will present a new entropy for the Doi-Onsager 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.
KITW01 7th September 2010
09:00 to 10:00
K Aoki Some decay problems of a collisionless gas: Numerical study
We investigate time-dependent 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 non-oscillatory decay. This is a joint work with T. Tsuji.
KITW01 7th September 2010
10:00 to 11:00
L Desvillettes 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 Vlasov-Fokker-Planck, or Vlasov-Boltzmann) equation is coupled with an equation of fluid mechanics (compressible or incompressible Euler; compressible or incompressible Navier-stokes; 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.
KITW01 7th September 2010
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 Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact 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.
KITW01 7th September 2010
14:00 to 15:00
Viscosity of bacterial suspensions: experiment and theory
Measurements of the shear viscosity in suspensions of swimming Bacillus subtilis in free-standing 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 self-propulsion 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, Three-dimensional model for the effective viscosity of bacterial suspensions, Phys. Rev. E 80, 041922 (2009)
KITW01 7th September 2010
15:30 to 16:10
A E Tzavaras 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 Fokker-Planck 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 Oldroyd-B models) and spurt phenomena or shear bands in that context. Also, that the long-time behavior of the sedimentating ow is approximated in a diusive scaling by the Keller-Segel model. (joint work with Ch. Helzel, U. Bochum and F. Otto, Leipzig).
KITW01 7th September 2010
16:10 to 16:50
Some polymeric fluid flow models: steady states and large-time convergence
We consider a dumbbell model for a dilute solution of polymers in a homogeneous fluid. In a micro-macro model, the incompressible Navier-Stokes equation for the fluid flow is coupled to a Fokker-Planck equation for the (microscipic) distribution of the polymeric chains. First we analyze the linear Fokker-Planck equation for Hookean dumbbells and in the case of finite extension nonlinear elasticity (FENE): steady states and large-time 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 long-time asymptotics for some polymeric fluid flow models, Comm. Math. Sc. 8, No. 3 (2010) 763-782. A. Arnold, C. Bardos, I. Catto: Stable steady states of a FENE-dumbbell model for polymeric fluid flows, preprint, 2010.
KITW01 7th September 2010
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.
KITW01 7th September 2010
17:30 to 18:00
J Haskovec 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 ring-shaped 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 Fokker-Planck 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.
KITW01 7th September 2010
18:00 to 18:30
Asymptotic dynamics of a population density: a model with a survival threshold
We study the long time-long range behavior of reaction diffusion equations with negative square-root reaction terms. In particular we investigate the exponential behavior of the solutions after a standard hyperbolic scaling. This leads to a Hamilton-Jacobi equality with an obstacle that depends on the solution itself. Our motivation comes from the so-called “tail problem” in population biology. We impose extra-mortality below a given survival threshold to avoid meaningless exponential tails. This is a joint work with G. Barles, B. Perthame and P. E. Souganidis.
KITW01 8th September 2010
09:00 to 10:00
Hydrodynamic limits, Knudsen layers and Numerical fluxes
KITW01 8th September 2010
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.
KITW01 8th September 2010
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 first-principles 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.
KITW01 8th September 2010
14:00 to 15:00
P Maini 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.
KITW01 8th September 2010
15:30 to 16:10
N Bournaveas 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).
KITW01 8th September 2010
16:10 to 16:50
V Calvez Kinetic models for bacterial chemotaxis
In this talk we discuss kinetic models for self-organization of cells. The framework is the Othmer-Dunbar-Alt model. It describes a velocity-jump process accounting for the motion of cells (e.g. bacteria) biased by a chemical signal. Several behaviours can be observed, ranging from blow-up in finite time to propagation of traveling waves. Blow-up 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).
KITW01 8th September 2010
16:50 to 17:30
Cross diffusion preventing blow up in the two-dimensional Keller-Segel model
We analyse the parabolic Keller-Segel system with an additional cross-diffusion term guaranteeing global-in-time 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 semi-discretisation 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 blow-up.
KITW01 8th September 2010
17:30 to 18:00
An integro-differential model to study evolution
The evolution of a population structured by a continuous phenotypic trait can be modeled by integro-differential 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.
KITW01 9th September 2010
09:00 to 10:00
Kinetic description and connectivity of old and new models of flocking
We introduce a new particle-based model for flocking and we show that, as with the Cucker-Smale model, flocking occurs when pairwise long range interactions are sufficiently strong. Next, we derive a Vlasov-type kinetic model for these particle models and we inquire about their time-asymptotic 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.
KITW01 9th September 2010
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.
KITW01 9th September 2010
11:30 to 12:30
P Souganidis 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.
KITW01 9th September 2010
13:30 to 14:10
S Fedotov Subdiffusion and nonlinear reaction-transport equations
The main aim of the talk is to discuss how to incorporate the nonlinear kinetic term into non-Markovian 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 reaction-transport systems. Applications include the transport of particles in spiny dendrites, the proliferation and migration dichotomy of the tumor cell invasion.
KITW01 9th September 2010
14:10 to 14:50
Wave propagation in non-homogeneous semilinear reaction-diffusion
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 reaction-diffusion equations. Respects in which the homogeneous case is non-generic when embedded within a heterogeneous setting will be highlighted.
KITW01 9th September 2010
15:30 to 16:00
Coupled Chemotaxis-Fluid Models
We consider coupled chemotaxis-fluid models aimed to describe swimming bacteria, which show bio-convective 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 global-in-time existence of solutions to the Cauchy problem in two and three space dimensions is established. Precisely, for the chemotaxis-Navier-Stokes 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 global-in-time existence of weak solutions for cell density with finite mass, first-order 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 global-in-time existence of weak solutions in two space dimensions.
KITW01 9th September 2010
16:00 to 16:30
LM Tine Existence and uniqness of solution to the Lifshitz-Slyozov system with monomers spatial diffusion
The standard Lifshitz-Slyozov model describes the evolution of a population of macro-particles or polymers immersed in a bath of monomers. It appears in such solution interaction phenomena between macro-particles 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.
KITW01 9th September 2010
16:30 to 17:30
Fokker-Planck models for Bose-Einstein particles
We study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion 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 large-time 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.
KITW01 10th September 2010
09:00 to 10:00
Fluid models of swarming behavior
Swarming biological systems (such as fish schools, insect swarms, mammalian herds) exhibit large scale spatio-temporal 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.
KITW01 10th September 2010
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 fluid-kinetic regions. We emphasize limitations presented by several standard schemes and focus our attention on a class of exponential Runge-Kutta 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.
KITW01 10th September 2010
11:30 to 12:30
A class of self-similar solutions for the Vlasov-Einstein system (joint work with A. Rendall)
In this talk I will describe a family of spherically symmetric self-similar solutions for the massless Einstein-Vlasov system. These solutions are supported in some particular hypersurfaces of the phase space and therefore, they are measure-like solutions, not fully dispersive.

The construction of these solutions relies in reformulating the problem as a four-dimensional dynamical systems problem that is studied in detail in a suitable perturbative limit.
KITW01 10th September 2010
14:00 to 14:40
Trend to the equilibrium for kinetic Fokker-Planck 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 transport-diffusive equations with periodic boundary conditions in the spatial variable. The diffusive part is given by the Laplace-Beltrami 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.
KITW01 10th September 2010
15:10 to 15:50
DSMC-fluid 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. 665-694, (2005).

[2] P.Degond, G. Dimarco, L. Mieussens., A Moving Interface Method for Dynamic Kinetic-fluid Coupling. J. Comp. Phys., Vol. 227, pp. 1176-1208, (2007).

[3] P.Degond, G. Dimarco, L. Mieussens., A Multiscale Kinetic-Fluid Solver with Dynamic Localization of Kinetic Effects. J. Comp. Phys., Vol. 229, pp.4907-4933, (2010).

[4] P. Degond, G. Dimarco, L. Pareschi, The Moment Guided Monte Carlo Method, To appear in International Journal for Numerical Methods in Fluids.
KITW01 10th September 2010
15:50 to 16:30
Aggregation-pattern due to repulsive-aggregating interaction potentials
We study non-local 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.
KITW01 10th September 2010
16:30 to 17:00
P Carcaud Study of two simplified kinetic models for evaporation in gravitation
KITW01 10th September 2010
17:00 to 17:30
Large time behavior of collisionless plasma
The motion of a collisionless plasma, a high-temperature, low-density, ionized gas, is described by the Vlasov-Maxwell equations. In the presence of large velocities, relativistic corrections are meaningful, and when magnetic effects are neglected this formally becomes the relativistic Vlasov-Poisson system. Similarly, if one takes the classical limit as the speed of light tends to infinity, one obtains the classical Vlasov-Poisson system. We study the long-time 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.
KIT 14th September 2010
14:00 to 14:45
Numerical exploration of a forward-backward diffusion equation
We analyze numerically a forward-backward diffusion equation with a cubic-like diffusion function, -emerging in the framework of phase transitions modeling- and its "entropy" formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we propose schemes for both the second and the third order equations, we discuss the analytical properties of their semi-discrete counter-parts and we compare the numerical results in the case of initial data of Riemann type, showing strengths and flaws of the two approaches, the main emphasis being given to the propagation of transition interfaces.
KIT 14th September 2010
14:50 to 15:35
Poiseuille and thermal transpiration flows of a highly rarefied gas
Poiseuille and thermal transpiration flows of a highly rarefied gas are investigated on the basis of the linearized Boltzmann equation, with a special interest in the overconcentration of molecules with a velocity parallel to the walls. An iterative approximation procedure with an explicit error estimate is presented, by which the structure of the over-concentration is clarified. A numerical computation on the basis of the procedure is performed for a hard-sphere molecular gas to construct a database that promptly gives the induced net mass flow for an arbitrary value of large Knudsen number. An asymptotic formula of the net mass flow is also presented for molecular models belonging to Grad's hard-potential. Finally, the resemblance of the profiles between the heat flow of the Poiseuille flow and the flow velocity of the thermal transpiration will be pointed out. This is a joint work with Hitoshi FUNAGANE.
KIT 21st September 2010
14:00 to 14:45
Lattice Boltzmann equation: what Do We Know and What Can We Do With It?
The primary purpose of this talk is to satisfy the curiosity of those who may be interested in the lattice Boltzmann equation (LBE): What is LBE? And what can it do for us?

Thus, I will first show how to derive the LBE from the continuous linearized Boltzmann equation. Through the derivation, some basic mathematical features of the LBE become apparent. First, it is a discrete moment system suitable for solving near incompressible (or low speed) flows and it can be related to the method of artificial compressibility. Second, it cannot be used to solve the continuous Boltzmann equation, because it cannot accurately model the higher order moments of the distribution function due to the constraint imposed by the symmetry of the underlying lattice. And third, the LBE can be related to finite-difference method; it is a second-order accurate solver for incompressible Navier-Stokes equation. To demonstrate the capability of the LBE, I will show a comparative study of the LBE and the pseudo-spectral (PS) method for direct numerical simulations (DNS) of the decaying turbulence in a three dimensional periodic cube. Not only some statistical quantities, but also the instantaneous flow fields (velocity and vorticity) are compared in details. Our study demonstrates that the results obtained by the LBE and PS methods agree very well when the flow is properly resolved in both methods. Our results indicate that the LBE requires twice as many grid points in each dimension as in the PS method. LBE simulations of complex flows (such as particulate suspensions, free-surface flows, and multi-component flow through porous media, etc.) will also be shown if time permits.

KIT 21st September 2010
15:00 to 15:45
S Rjasanow Numerics of the inelastic Boltzmann equation
In the present lecture we give an overview of the analytical and numerical properties of the spatially homogeneous granular Boltzmann equation. The presentation is based on the three articles. In the first article [1], we consider the uniformly heated spatially homogeneous granular Boltzmann equation. A new stochastic numerical algorithm for this problem is presented and tested using analytical relaxation of the temperature. The tails of the steady state distribution, which are overpopulated for the steady state solutions of the granular Boltzmann equation, are computed using this algorithm and the results are compared with the available analytical information. In the second paper [2], we deal again with the same equation and consider the DSMC-error due to splitting technique using the time step Delta_t. This equation provides the possibility to see the Delta_t error of the first order when using standard splitting technique and of the second order for Strang?s splitting method. In contrast, a new direct simulation method recently introduced in [1] is "exact", i.e. the error is a function of the number of particles n and of the number of independent ensembles N_rep. Thus this method can be seen as a "correct" generalization of the classical DSMC for the inelastic Boltzmann equation. Finally, in [3], we consider an inelastic gas in a host medium. The numerical algorithm is tested by the use of the analytical relaxation of the momentum and of the temperature.
References

[1] I. M. Gamba, S. Rjasanow, and W. Wagner. Direct simulation of the uniformly heated granular Boltzmann equation. Math. Comput. Modelling , 42(5-6):683?700, 2005.

[2] S. Rjasanow and W. Wagner. Time splitting error in DSMC schemes for the inelastic Boltzmann equation. SIAM J. Numerical Anal., 45(1):54? 67, 2007.

[3] M. Bisi, S. Rjasanow and G. Spiga. A numerical study of the granular gas in a host medium. Preprint NI10051 of the Isaac Newton Institute for Mathematical Sciences, 2010.

KIT 28th September 2010
14:00 to 14:45
Hilbert sixth problem
Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.
KIT 28th September 2010
15:00 to 15:45
Linear Boltzmann equation and some Dirichlet series
It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta function) can be presented as a class of solutions of the Fourier transformed spatially homogeneous linear Boltzmann equation with a special Maxwell type collision kernel. The proof uses an explicit integral representation of solutions to the Cauchy problem for the Boltzmann equation. Possible applications to the theory of Dirichlet series are briefly discussed. The talk is based on joint paper with Irene Gamba.
KIT 4th October 2010
14:00 to 14:45
Series of lectures on Landau damping I
KIT 4th October 2010
15:15 to 16:00
Series of lectures on Landau damping II
KIT 5th October 2010
14:00 to 14:45
Interaction dynamics of singular wave fronts computed by particle methods
Some of the most impressive singular wave fronts seen in Nature are the transbasin oceanic internal waves, which may be observed from a space shuttle, as they propagate and interact with each other. The characteristic feature of these strongly nonlinear waves is that they reconnect whenever any two of them collide transversely. The dynami cs of these internal wave fronts is governed by the so-called EPDiff equation, which, in particular, coincides with the dispersionless case of the Camassa-Holm (CH) equation for shallow water in one- and two-dimensions.

Typical weak solutions of the EPDiff equation are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. The equation admits solutions that are nonlinear superpositions of traveling waves and troughs that have a discontinuity in the first derivative at their peaks and therefore are called peakons. Capturing these solutions numerically is a challenging task especially when a peakon-antipeakon interaction needs to be resolved.

In this talk, I will present a particle method for the numerical simulation and investigation of solitary wave structures of the EPDiff equation in one and two dimensions. I will show that the discretization of the EPDiff by means of the particle method preserves the basic Hamiltonian, the weak and variational structure of the original problem, and respects the conservation laws associated with symmetry under the Euclidean group. I will also present a convergence analysis of the proposed particle method in 1-D.

Finally, I will demonstrate the performance of the particle methods on a number of numerical examples in both one and two dimensions. The numerical results illustrate that the particle method has superior features and represent huge computational savings when the initial data of interest lies on a submanifold. The method can also be effectively implemented in straightforward fashion in a parallel computing environment for arbitrary initial data.

KIT 5th October 2010
15:00 to 15:45
Gradient flow scheme for nonlinear fourth order equations
Evolution equations with an underlying gradient flow structure have since long been of special interest in analysis and mathematical physics. In particular, transport equations that allow for a variational formulation with respect to the L2-Wasserstein metric have attracted a lot of attention recently. The gradient flow formulation gives rise to a natural semi-discretization in time of the evolution by means of the minimizing movement scheme (see, e.g. [1]), which constitutes a time-discrete minimization problem for the (sum of kinetic and potential) energy. On the other hand, nonlinear diffusion equations of fourth (and higher) order have become increasingly important in pure and applied mathematics. Many of them have been interpreted as gradient flows with respect to some metric structure.

When it comes to solve equations of a gradient flow type numerically, it is natural to ask for appropriate schemes that respect the equation's special structure in some way. We propose a fully discrete variant of the minimizing movement scheme for numerical solution of the nonlinear fourth order Derrida-Lebowitz-Speer-Spohn equation (cf. [2]). Our method is iterative and consists of an outer and an inner loop. In each time step of the outer loop a constrained quadratic optimization problem for the Fisher information is solved on a finite-dimensional space of ansatz functions. These subproblems are solved iteratively in the inner loop by applying Newton's method to the optimality system, leading to a sequential quadratic programming method. In result, we obtain a fully practical numerical scheme for a non-linear fourth order equation that respects its Wasserstein gradient flow structure.

Joint work with Daniel Matthes and Pina Milisic, see [3].

References [1] L. Ambrosio and G. Savaré. Gradient flows of probability measures. In: Handbook of Evolution Equations, Dafermos, C. and Feireisl, E. (eds.), Vol. 3, pp. 1-136, Elsevier, 2006. [2] A. Jüngel and D. Matthes. The Derrida-Lebowitz-Speer-Spohn equation: existence, nonuniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39(6), 1996-2015, 2008. [3] B. Düring, D. Matthes, and J.-P. Milisic. A gradient flow scheme for nonlinear fourth order equations. Discrete Contin. Dyn. Syst. Ser. B 14(3) (2010), 935-959.

KIT 11th October 2010
14:00 to 14:45
Series of lectures on Landau damping III
KIT 11th October 2010
15:15 to 16:00
Series of lectures on Landau damping IV
KIT 12th October 2010
14:00 to 14:45
I Gamba Convolution inequalities for the Boltzmann operator and the inhomogeneous BTE with soft potentials for initial data near Maxwellian states
KIT 12th October 2010
15:00 to 15:45
V Panferov On kinetic and hydrodynamic descriptions of the Vicsek dynamics
The Viscek model (1995) has been a prototype for many studies of self-organization and collective behavior in animal groups. The basic formulation consists of a system of N paricles moving on the plane (or in space) with constant speed and adjusting their velocities at discrete time instants to the local averages, while subject to random perturbations (noise). We consider a related purely deterministic system in which alignment effects are combined with local repulsive force interactions, and which has been used previously to model large scale behavior of pelagic fish schools. Scaling the parameters of the model in a particular way such that the discrete time step approaches zero and the number of particles N approaches infinity, we find nontrivial hydrodynamic regimes in which a reduced description of the system through the macroscopic density and the average angle becomes possible. We present a systematic procedure of deriving the hydrodynamic equations and numerical results on comparison of particle and continuum (PDE) solutions.
KIT 18th October 2010
14:00 to 14:45
Series of lectures on Landau damping V
KIT 18th October 2010
15:15 to 16:00
Series of lectures on Landau damping VI
KIT 19th October 2010
14:00 to 14:45
W Strauss Stability theory in a collisionless plasma
For a collisionless plasma that is modeled by the relativistic Vlasov-Maxwell system, many equilibria are stable but many others are unstable. In this talk, presenting joint work with Zhiwu Lin, I will consider axisymmetric equilibria of the form f(e, p) that are decreasing in the particle energy e and also depend on the particle angular momentum p. Then a necessary and sufficient condition for linear stability is the positivity of a certain linear operator L^0. This operator L^0 is much less complicated than the generator of the full linearized system. It has an interesting non-local term that can definitely affect its positivity. There is a similar reduction in the simpler case of 1.5 dimensional symmetry. For the important example of a purely magnetic equilibrium, explicit conditions for the linear/nonlinear stability/instability are obtained.
KIT 19th October 2010
15:00 to 15:45
F Bolley Mean field limit of stochastic particle systems
We consider a large system of particles in R^d, evolving according to coupled stochastic differential equations, motivated by recenty studied swarming models. In the limit when the size of the system goes to infinity, its behaviour is described by an mean field kinetic PDE.

The issue is to justify the limit and make it quantitative. This is a joint work with José A. Cañizo and José A. Carrillo.

KIT 25th October 2010
14:00 to 14:45
Series of lectures on Landau damping VII
KIT 25th October 2010
15:15 to 16:00
Series of lectures on Landau damping VIII
KIT 26th October 2010
14:00 to 14:45
A completely integrable toy model of nonlinear Schrodinger equations without dispersion
I shall discuss the cubic Szego equation which is the Hamiltonian evolution associated to the L^4 norm on the Hardy space of the circle, and explain why it is a toy model for NLS without dispersion. I shall prove that this evolution admits a Lax pair, and use this structure to solve explicitely the Cauchy problem through some inverse spectral problem, and discuss various stability questions.

This is a joint work with Sandrine Grellier (Orleans)

KIT 26th October 2010
15:00 to 15:55
P Laurencot A stochastic min-driven coalescence process and its hydrodynamical limit
A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalised version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models. (joint work with Anne-Laure Basdevant (Paris X), James R. Norris (Cambridge), Cl\'ement Rau (Toulouse))
KIT 1st November 2010
14:00 to 14:45
Series of lectures on Landau damping IX
KIT 1st November 2010
15:15 to 16:00
Series of lectures on Landau damping X
KIT 2nd November 2010
14:00 to 14:45
A Nouri On kinetic equations describing tokamak plasmas
In the core of a tokamak, the evolution of the ions distribution function can be described by a kinetic transport equation whereas the electrons are assumed to be at thermodynamic equlibrium and the electrical potential is givne by the electro-neutrality equation. Moreover, gyroaverages are performed in the directions perpendicular to the magnetic field. This induces different properties of the ion distribution function in the directions parallel and perpendicular to the magnetic field. In this talk we will stress them when deriving existence and uniqueness of stationary solutions, as well as well-posedness of a diffusive gyrokinetic model.
KIT 2nd November 2010
15:00 to 15:45
S Mischler Factorisation for non-symmetric operators and exponential H-theorems
We present a factorization method for estimating resolvents of non-symmetric operators in Banach or Hilbert spaces in terms of estimates in another (typically smaller) ``reference'' space. This applies to a class of operators writing as a ``regularizing'' part (in a broad sense) plus a dissipative part. Then in the Hilbert case we combine this factorization approach with an abstract Plancherel identity on the resolvent into a method for enlarging the functional space of decay estimates on semigroups. In the Banach case, we prove the same result however with some loss on the norm. We then apply these functional analysis approach to several PDEs: the Fokker-Planck and kinetic Fokker-Planck equations, the linear scattering Boltzmann equation in the torus, and, most importantly the linearized Boltzmann equation in the torus (at the price of extra specific work in the latter case). In addition to the abstract method in itself, the main outcome of the paper is indeed the first proof of exponential decay towards global equilibrium (e.g. in terms of the relative entropy) for the full Boltzmann equation for hard spheres, conditionnally to some smoothness and (polynomial) moment estimates. This improves on the result in [Desvillettes-Villani, Invent. Math., 2005] where the rate was ``almost exponential'', that is polynomial with exponent as high as wanted, and solves a long-standing conjecture about the rate of decay in the H-theorem for the nonlinear Boltzmann equation, see for instance [Cercignani, Arch. Mech, 1982] and [Rezakhanlou-Villani, Lecture Notes Springer, 2001].
KIT 15th November 2010
14:00 to 15:00
Particle systems and the hydro-dynamical limit of the Ginzburg-Landau model
KIT 15th November 2010
17:00 to 18:00
Entropy and H theorem: The mathematical legacy of Ludwig Boltzmann
This talk is devoted to the presentation of some of the most important concepts in statistical mechanics, including Boltzmann's statistical entropy, the notion of macroscopic irreversibility and molecular chaos, and the Boltzmann equation.
KIT 16th November 2010
14:00 to 14:45
Ghost effect by curvature
We consider the Boltzmann equation for a gas between two rotating coaxial cylinders, and we study the behaviour of the gas in the limit in which the Knudsen number k and the inverse of radius (the curvature) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The Mach number goes to zero as k_a, a
KIT 16th November 2010
15:00 to 15:45
'The Hughes' model for pedestrian flow
I will present a recent result in collaboration with Peter A Markowich, Jan F Pletschmann, and Marie T Wolfram (DAMTP, University of Cambridge) about the mathematical theory of the Hughes' model for the flow of pedestrians (cf. Hughes 2002), which consists of a continuity equation with logistic mobility coupled with an eikonal equation for the potential describing the common sense of the task.

We consider an approximation of such model in which the eikonal equation is replaced by an elliptic approximation.

For such an approximated model we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kruzkov, in which the boundary conditions are posed following the approach by Bardos et al. We use BV estimates on the density and stability estimates on the potential in order to prove uniqueness.

Moreover, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behaviour of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.

KITW05 22nd November 2010
09:35 to 10:25
On some multilocus migration-selection models
The dynamics and equilibrium structure of a general deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is ergodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. For two are particularly interesting limiting cases, weak or strong migration, global (regular or singular) perturbation results establish generic convergence of trajectories and allow for a detailed study of the equilibrium structure. Applications to the problem of the maintenance of genetic variation in structured populations are briefly outlined.
KITW05 22nd November 2010
11:00 to 11:50
Bifurcation problems for structured population dynamics models
This presentation is devoted to bifurcation problems for some classes of PDE arising in the context of population dynamics. The main difficulty in such a context is to understand the dynamical properties of a PDE with non-linear and non-local boundary conditions.

A typical class of examples is the so called age structured models. Age structured models have been well understood in terms of existence, uniqueness, and stability of equilibria since the 80's. Nevertheless, up to recently, the bifurcation properties of the semiflow generated by such a system has been only poorly understood.

In this presentation, we will start with some results about existence and smoothness of the center mainfold, and we will present some general Hopf bifurcation results applying to age structured models. Then we will turn to normal theory in such a context. The point here is to obtain formula to compute the first order terms of the Taylor expansion of the reduced system.

KITW05 22nd November 2010
11:50 to 12:40
Asymptotic spreading in general heterogeneous media
We will present in this talk propagation properties for the solutions of the heterogeneous Fisher-KPP equation $$\partial_{t} u - \partial_{xx}u=\mu (t,x) u(1-u)$$ where $\mu$ is only assumed to be uniformly continuous and bounded in $(t,x)$, for initial data with compact support. Using homogenization techniques, we construct two speeds $\overline{w}$ and $\underline{w}$ such that $\lim_{t\to+\infty}u(t,x+wt) = 0$ if $w>\overline{w}$ and $\lim_{t\to+\infty} u(t,x+wt)=1$ if $w
KITW05 22nd November 2010
14:30 to 15:20
J Polechova Adaptation in continuous populations with migration and genetic drift
What are the intrinsic causes of the limits to adaptation throughout a species' range and at its edge? I give an introduction to the problem, and provide some results on the evolution of clines in allele frequencies due to drift, selection and migration:

Random genetic drift shifts clines from side to side, alters their width, and distorts their shape. The wobbling in position makes the expected cline wider. However, locally, drift drives alleles towards fixation, so individual clines can often be narrower. I show that the relation between the deterministic cline width, expected cline width, and width of the expected cline is driven by the average standardized variance of local allele frequencies.

KITW05 22nd November 2010
16:00 to 16:50
The effect of selection on genealogies, and a near-critical system of branching Brownian motions
Consider the following model for the evolution of a population undergoing natural selection. At each generation, individuals give a fixed or random number of offsprings, whose fitness is perturbed by some independent additive noise. Selection maintains the population size fixed by selecting only the fittest individuals. Physicists have recently made some very precise predictions for the genealogy of this population in the limit of inifinite population size : namely, the characteristic timescale is (log N)^3 generations, and when measured in these units, genealogical trees are described by the so-called Bolthausen-Sznitman coalescent. These predictions build on a non-rigorous analysis of stochastic effects in front propagation of KPP equations. We report on some recent progress on this issue (joint work with J. Berestycki and J. Schweinsberg), where this conjecture is shown to hold for a slightly simpler model of branching Brownian motions with a suitable cutoff which keeps the population size approximately fixed.
KIT 23rd November 2010
14:00 to 14:45
I Gasser Traffic dynamics induced by bottleneck
We study the dynamics of traffic on a circular road by using a simple car following model. In case of no bottleneck the situation is well known. Especially the loss of stability for higher traffic density well understood. Introducing a bottleneck the situation becomes more complex and a completely different approach is needed. Doing so the bifurcations and the dynamics of the bottleneck case can be studied and many new interesting phenomena can be observed.
KIT 23rd November 2010
15:00 to 15:45
M Gualdani Decay estimates towards equilibrium for a quantum kinetic equation
KIT 29th November 2010
14:00 to 15:00
W Wagner Kinetic equations and Markov jump processes
The first part of the talk is an introduction. The Boltzmann equation, Smoluchowski's coagulation equation and the multiple fragmentation equation are considered and related stochastic particle models are introduced. In the second part of the talk, a particular phase transition in fragmentation models is studied. This transition is related to the formation of infinitely small particles. It corresponds to the existence of non-conservative solutions of the kinetic equation and to explosion (fast accumulation of jumps) in the stochastic model.
KIT 30th November 2010
14:00 to 14:45
JA Carrillo Nonlinear integrate and fire neuron models: analysis and numerics
Abstract: Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter. This is a work in collaboration with M. J. Cáceres and B. Perthame.
KIT 30th November 2010
15:00 to 15:45
Nonlinear integrate and fire neuron models, part II: existence, uniqueness, and asymptotics
We continue the study of Nonlinear Noisy Leaky Integrate and Fire models for neurons. In particular, we show global existence and uniqueness of the solution for the inhibitory case, together with some comments on asymptotic decay. The main idea is to reduce the equation to a Stefan-like problem with a moving free boundary and non-standard right hand side. This is joint work with J. Carrillo, M. Gualdani and M. Schonbek.
KITW03 13th December 2010
10:00 to 11:00
S Jin An Eulerian surface hopping method for the Schrödinger equation with conical crossings
In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schrödinger equation with conical crossings in the Eulerian formulation. The approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau-Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. While a Lagrangian particle method requires the increase in time of the particle numbers, or a large number of statistical samples in a Monte Carlo setting, the advantage of an Eulerian method is that it relies on fixed number of partial differential equations with a uniform in time computational accuracy. We prove the positivity and $l^{1}$-stability and illustrate by several numerical examples the validity and accuracy of the proposed method.
KITW03 13th December 2010
11:30 to 12:30
T Paul Strong semiclassical approximation for the Hartree dynamics
We present some recent results concerning the semiclassical approximation for the Hartree equation in strong topoogly.
KITW03 13th December 2010
14:00 to 15:00
Thermic effects in Hartree systems
The goal of this lecture is to consider solutions of the Hartree and Hartree-Fock systems, both in attractive and repulsive cases, corresponding to non-zero temperatures. Such solutions are computed by minimizing a free energy functional, which can be proved to be bounded from below using interpolation inequalities. These Gagliardo-Nirenberg interpolation inequalities are equivalent to Lieb-Thirring inequalities. Various effects due to the temperature are charaterized, which also depend on the entropy generating function.

[1] J. Dolbeault, P. Felmer, M. Loss, and E. Paturel. Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems. J. Funct. Anal., 238 (1): 193-220, 2006

[2] J. Dolbeault, P. Felmer, and J. Mayorga-Zambrano. Compactness properties for trace-class operators and applications to quantum mechanics. Monatshefte für Mathematik, 155 (1): 43-66, 2008

[3] J. Dolbeault, P. Felmer, and M. Lewin. Orbitally stable states in generalized Hartree-Fock theory. Mathematical Models and Methods in Applied Sciences, 19 (3): 347-367, 2009.

[4] G. L. Aki, J. Dolbeault and C. Sparber. Thermal effects in gravitational Hartree systems. Preprint
KITW03 13th December 2010
15:30 to 16:30
Remarks on Fermi-Dirac and Bose-Einstein quantum Boltzmann type equations
We show some complementary results which extend slightly the works of X. Lu M. Escobedo, S. Mischler and M.-A. Valle related to some quantum Boltzmann models, and more precisely Fermi-Dirac and Bose-Einstein particles.
KITW03 13th December 2010
16:30 to 17:30
Tailored finite point method for wave propagation
In this talk, we propose a tailored-finite-point method for the numerical simulation of wave propagation with high wave numbers in heterogeneous media. It is a joint work with Prof. Han and Dr. Yang.
KITW03 14th December 2010
09:00 to 10:00
W Bao Quantized vortex stability and dynamics in superfluidity and superconductivity
In this talk, I will review our recent work on quantized vortex stability and dynamics in Ginzburg-Landau-Schroedinger and nonlinear wave equation for modeling superfluidity and superconductivity as well as nonlinear optics. The reduced dynamic laws for quantized vortex interaction are reviewed and solved analytically in several cases. Direct numerical simulation results for Ginzburg-Landau-Schroedinger and nonlinear wave equations are reported for quantized vortex dynamics and they are compared with those from the reduced dynamics laws.

References:

[1] Y. Zhang, W. Bao and Q. Du, The Dynamics and Interaction of Quantized Vortices in Ginzburg-Landau-Schroedinger equations, SIAM J. Appl. Math., Vol. 67, No. 6, pp. 1740-1775, 2007

[2] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrodinger equation, Eur. J. Appl. Math., Vol. 18, pp. 607-630, 2007.

[3] W. Bao, Q. Du and Y. Zhang, Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numericalcomputation, SIAM J. Appl. Math., Vol. 66 , No. 3, pp. 758-786, 2006.
KITW03 14th December 2010
10:00 to 11:00
NRxx Method for Boltzmann-BGK Equation
We introduce a numerical method for solving Grad's moment equations or regularized moment equations for arbitrary order of moments. In our algorithm, we do not explicitly need the moment equations. Instead, we directly start from the Boltzmann equation and perform Grad's moment method. We define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions. For the collision term, the BGK model is adopted so that the production step can be done trivially based on the Hermite expansion. To regularize the system, we apply the Maxwell iteration to the infinite moment system and determine the magnitude of each moment with respect to the Knudsen number. After that, we obtain the approximation of high order moments and close the moment systems by dropping some high-order terms. Linearization is then performed to obtain a very simple regularization term, thus it is very convenient for numerical implementation.
KITW03 14th December 2010
11:30 to 12:30
A Juengel Analysis of diffusive quantum fluid models
Quantum fluid models have been recently derived by Degond, Mehats, and Ringhofer from the Wigner-BGK equation by a moment method with a quantum Maxwellian closure. In the O(eps^4) approximation, where eps is the scaled Planck constant, this leads to local quantum diffusion or quantum hydrodynamic equations. In this talk, we present recent results on the global existence and long-time decay of solutions of these models.

First, we consider quantum diffusion models containing highly nonlinear fourth-order or sixth-order differential operators. The existence results are obtained from a priori estimates using entropy dissipation methods. Second, a quantum Navier-Stokes model, derived by Brull and Mehats, will be analyzed. This system contains nonlinear third-order derivatives and a density-depending viscosity. The key idea of the mathematical analysis is the reformulation of the system in terms of a new "osmotic velocity" variable, leading to a viscous quantum hydrodynamic model. Surprisingly, this variable has been also successfully employed by Bresch and Desjardins in (non-quantum) viscous Korteweg models.
KITW03 14th December 2010
14:00 to 15:00
On the moment problem for quantum hydrodynamics
We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.
KITW03 14th December 2010
15:30 to 16:30
Global Ray Tracing
It is long argued that the classical geometric optics (GO) method based on the WKB ansatz suffers from the existence of caustics, thus the GO has to be modified in some manner such as the Gaussian beam approach does.

In this talk, I will show that a simple remedy exists.
KITW03 15th December 2010
09:00 to 10:00
C Lasser Semiclassics beyond the Egorov theorem
KITW03 15th December 2010
10:00 to 11:00
Continuations of the nonlinear Schrodinger solutions beyond the singularity
The continuation of NLS solutions beyond the singularity has been an open problem for many years. In this talk I will present several novel approaches to this problem, and discuss their mathematical and physical consequences.
KITW03 15th December 2010
11:30 to 12:30
H Wu Bloch Decomposition-Based Gaussian Beam Method for the Schrödinger equation with Periodic Potentials
KITW03 15th December 2010
14:00 to 15:00
R Carles Nonlinear coherent states and Ehrenfest time for Schrodinger equation
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.
KITW03 15th December 2010
15:30 to 16:30
Control of Quantum Systems
KITW03 16th December 2010
09:00 to 10:00
On Bohmian measures and their classical limit
We review several recent results on a class of mono-kinetic phase space measures arising from the Bohmian interpretation of quantum mechanics. In particular we shall be concerned with the corresponding classical limit in comparison to the well known theory of Wigner measures. This is a joint work with P. Markowich and T. Paul.
KITW03 16th December 2010
10:00 to 11:00
A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime
Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision term. A class of asymptotic preserving schemes was introduced in [5] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose- Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi- Dirac function). In this paper, we propose to penalize the quantum collision term by a 'classical' BGK operator instead of the quantum one. This is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are present to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.
KITW03 16th December 2010
11:30 to 12:30
P Antonelli Scattering and asymptotic completness to the Schrödinger equation with critical Nonlinear and Hartree potentials
KITW03 16th December 2010
14:00 to 15:00
Blow-up Conditions for a System of Nonlinear Schrödinger Equations
We are considering a system of two coupled Schrödinger equations with focusing and defocusing nonlinearities. In this context we observe collapse of only one or of both components, respectively, depending on the strength of the nonlinearities. We deal with some conditions for the blow-up. Finallly, we present some numerical simulation results.
KITW03 16th December 2010
15:30 to 16:30
Semiclassical limit of nonlinear Schrödinger and Davey-Stewartson equations
The semiclassical limit of nonlinear Schrödinger and Davey-Stewartson equations is a small dispersion limit of purely dispersive PDEs. In the vicinity of shocks of the corresponding dispersionless equations, the solutions develop a zone of rapid modulated oscillations. The case of the focusing equations is especially interesting since the dispersionless equations as well as Whitham's averaged equations are elliptic. We present an asymptotic description of the breakup in the 1-dimensional case and a numerical study of the higher dimensional cases.
KITW03 17th December 2010
09:00 to 10:00
JC Saut From Gross-Pitaevskii to KdV and KP
We will survey recent results and open questions on the long wave limit of the Gross-Pitaevskii equation (cubic defocusing nonlinear Schrödinger equation with finite Ginzburg-Landau energy) in the so-called transonic regime.
KITW03 17th December 2010
10:00 to 11:00
Dynamical modelling of nonequilibrium condensates
Quasiparticles in semiconductors --- such as microcavity polaritons --- can form condensates in which the steady-state density profile is set by the balance of pumping and decay. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation. By taking account of the polarization degree of freedom for such condensates, and considering the effects of an applied magnetic field, it is possible to study the interplay between polarization dynamics, and the spatial structure of the pumped decaying condensate. Interactions between the spin components can influence the dynamics of vortices; produce stable complexes of vortices and rarefaction pulses with both co- and counter-rotating polarizations; and increase the range of possible limit cycles for the polarization dynamics, with different attractors displaying different spatial structures.
KITW03 17th December 2010
11:30 to 12:30
M Hintermüller Optimal control of the nonlinear Schrödinger equation
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons