The problems examined in these lectures concern asymptotic techniques and detailed quantitative properties close to global equilibrium in classical kinetic theory. For clarity of presentation, the discussion will mainly be centered on a particular two-rolls model problem for the Boltzmann equation and hard forces, with the understanding that such a program can be applied in many other contexts for single and multi-component gases. The topics include asymptotic expansions, a priori estimates, existence results, fluid dynamic limits, Taylor-Couette bifurcation and stability questions.

Program:

1. Common background and introduction of a prototype configuration: the particle, kinetic, and fluid modelling levels; the Boltzmann equation; a particular configuration and some significant boundary value problems; asymptotic expansions
2. Fluid dynamic and non fluid dynamic stationary a priori estimates: a linear existence result; non-hydrodynamic a priori estimates; comparison of estimates in various norms; hydrodynamic a priori estimates
3. Solutions to non-linear stationary kinetic problems: the rest term equation; contraction mappings controlled by these a priori estimates; bifurcations and fluid limits
4. On stability of stationary solutions: the stability problem; the solution program in the time-dependent case; long time behaviour
5. On positivity: a positivity preserving equation; on the positivity of the stationary Boltzmann equation; summary and comments

The main goal of this course is to give an overview of the recent ideas of contractivity of Wasserstein and Fourier-type distances applied to the asymptotic behavior of nonlinear diffusion, scalar conservation laws and granular flow models based on kinetic equations with inelastic interactions.

Program:

1. Basics of the Wasserstein distance & Model equations: nonlinear diffusions and homogeneous inelastic kinetic equations; description of models and main formal properties of their solutions; contractivity for 1-D models: consequences on qualitative properties of the solutions.
2. Gradient flows for the euclidean Wasserstein distance: asymptotic behavior for nonlinear diffusions and simplified homogeneous inelastic kinetic models; formal ideas behind gradient flows; contractivity in several dimensions; quantified displacement convexity and decay rates; entropy method versus mass transportation method for large time asymptotics.
3. Large time asymptotics for general nonlinear diffusion equations and scalar conservation laws: asymptotic profiles and identification of universal behavior; applications to the large time asymptotics of scalar conservation laws with integrable bounded positive initial data and decay rates towards N-waves.
4. Contractivity for the inelastic Boltzmann equation: Wassertein and Fourier-based metrics: conditions for being "equivalent"; contractivity of Wasserstein and Fourier-based distances for the inelastic homogeneous Boltzmann operator in the maxwellian approximation; existence, uniqueness, nonlinear asymptotic stability and decay rates for diffusive equilibria and homogeneous cooling states.

Program:

1. Generalized Kolmogorov-Avrami Models and their use in Physical Chemistry: the temporal dynamics of crystallization as described by probabilistic and deterministic models; in particular, partial crystallization in the presence of an amorphous side product will be discussed, a case not covered by the classical theory (tools used for this lecture are elementary probability theory, Voronoi diagrams and calculus).
2. Genetic Oscillators and their role in ultradian and circadian rhythms: definition and properties of transcriptional-translational oscillators, in short TTOs; their theoretical interplay in cell biology, and their possible role in circadian rhythms (tools used in this lecture include stochastic differential equations, time averaging and some gene biology).
3. Microscopic, kinetic and macroscopic multilane traffic dynamics: emphasis will be placed on Fokker-Planck type models for multilane traffic flow, but microscopic and macroscopic models will be introduced. It will be shown how fundamental diagrams are computed from the Fokker-Planck type models, and under what conditions they will be multivalued. Various applied questions related to traffic flow will be discussed, and local relaxation to equilibria states will be investigated via entropy methods.
4. An application of the Boltzmann equation for inelastic hard spheres in a turbulent background, or initial boundary value problems for the Boltzmann-Fokker-Planck equation: this question will be discussed with the application of ``coal dust'' in an incinerator in mind (students will be encouraged to assist with the modeling, think about reasonable boundary conditions, and suggest tools for an analytical treatment).

Motivated by the study of the asymptotic behavior of the Boltzmann equation, and by other applications in physics (e.g. stochastic forcing of partial differential equations, or modelling of heat conduction), the problem of convergence to equilibrium for degenerate equations recently got a lot of attention. Hypocoercivity, which is to coercivity what hypoellipticity is to ellipticity, seems to be a key concept in this problem. In the course the motivations of the theory and the state of the art will be reviewed.



Short Course on Microflows:
Kinetic Models for the Study of the Behavior of Fluids in Micro/Nano Electromechanical Systems
Carlo Cercignani

An area of application of kinetic theory has emerged in the last few years. Small size machines, called micromachines or nanomachines, are being designed and built. Their typical sizes range from a fraction of micron to a few millimeters. Rarefied flow phenomena that are more or less laboratory curiosities in machines of more usual size can form the basis of important systems in the micromechanical domain. In fact, rarefied gas flows occur in many micro-electro-mechanical systems (MEMS), such as actuators, microturbines, gas chromatographs, and micro air vehicles (MAVs). A correct prediction of these flows is important to design and develop MEMS. Nanoscale design occurs for computer components as well and is no longer limited to chip technology but extends to mechanical devices as well. In a modern disk drive, the read/write head floats at distances of the order of 50 nm above the surface of the spinning platter. The prediction of the pressure distribution in air is a crucial design calculation.

The program of the short course is organized as follows:

1. The linearized Boltzmann and its justification
2. Kinetic models
3. Plane Couette and Poiseuille flow
4. Transformation into an integral equation
5. Variational methods
6. How to calculate pressure in MEMS and NEMS: the modified Reynolds equation
7. Flow in a microchannel

Posters

• V. Bagland, B. Wennberg, Y. Wondmagegne, Stationary states for the non-cutoff Kac equation with a Gaussian thermostat
• C. Baranger, L. Boudin, J. Mathiaud, S. Pieraccini, Simulation platform for collisions kernels by DSMC method
• N. Bernhoff, A.V. Bobylev, Shock waves for the discrete Boltzmann equation
• M. Bisi, Reactive gases in a host medium: kinetic model and macroscopic equations
• Y. Chen, Smoothness of the solutions of generalized Landau equations
• M. Groppi, A. Aimi, M. Diligenti, G. Spiga, C. Guardasoni, On the BGK approximation of reactive flows: theoretical and numerical aspects
• J. Haskovec, C. Schmeiser, Diffusive limit of a kinetic model for cometary flows
• C. Manzini, G. Frosali, Compressed Chapman-Enskog analysis of a quantum transport model
• D. Matthes, J.P. Milisic, New quantum hydrodynamic equations for semiconductors
• C. Negulescu, A hybrid approach for the simulation of nanoscale MOSFETs: fully quantum models linked with quantum/classical adiabatic ones
• M. Vinerean, A. Bobylev, Discrete kinetic models with given conservation laws and their construction
• H. Yoshida, K. Aoki, Cylindrical Couette flows of a vapor-gas mixture: a ghost effect in the continuum limit

List of participants

Aoki Kazuo	Kyoto University, Japan	aoki@aero.mbox.media.kyoto-u.ac.jp
Arkeryd Leif	Chalmers University of Technology, Goteborg, Sweden	arkeryd@math.chalmers.se
Arsenio Diogo	Courant Institute, New York, USA	arsenio@cims.nyu.edu
Bagland Veronique	Universite' Paul Sabatier, Toulouse, France	bagland@mip.ups-tlse.fr
Baranger Celine	CEA, Paris, France	Celine.Baranger@cmla.ens-cachan.fr
Barletti Luigi	Universita' di Firenze, Italia	barletti@math.unifi.it
Belleni-Morante Aldo	Universita' di Firenze, Italia	abelleni@dicea.unifi.it
Bernhoff Niclas	University of Karlstad, Sweden	niclas.bernhoff@kau.se
Bisi Marzia	Universita' di Parma, Italia	marzia.bisi@unipr.it
Bobylev Alexander	University of Karlstad, Sweden	alexander.bobylev@kau.se
Borgioli Giovanni	Universita' di Firenze, Italia	giovanni.borgioli@unifi.it
Boudin Laurent	Universite' Paris VI, France	boudin@ann.jussieu.fr
Carrillo Jose'	Universitat Autonoma Barcelona, Espana	carrillo@mat.uab.es
Cavalli Fausto	Universita' di Milano, Italia	heyrou@tiscali.it
Cercignani Carlo	Politecnico di Milano, Italia	carcer@mate.polimi.it
Chen Yemin	Ecole Normale Superieure Cachan, France	yemin.chen@gmail.com
Chunjin Lin	Universite de Lille, France	chunjin.lin@math.univ-lille1.fr
Conforto Fiammetta	Universita' di Messina, Italia	fiamma@unime.it
Cordier Stephane	Universite' de Orleans, France	stephane.cordier@math.cnrs.fr
Desvillettes Laurent	Ecole Normale Superieure Cachan, France	Laurent.Desvillettes@cmla.ens-cachan.fr
Dimarco Giacomo	Universita' di Ferrara, italia	dmrgcm@unife.it
Filbet Francis	Universite' Paul Sabatier, Toulouse, France	filbet@mip.ups-tlse.fr
Frezzotti Aldo	Politecnico di Milano, Italia	aldo.frezzotti@polimi.it
Frosali Giovanni	Universita' di Firenze, Italia	frosali@dma.unifi.it
Furioli Giulia	Universita' di Bergamo, Italia	gfurioli@unibg.it
Groppi Maria	Universita' di Parma, Italia	maria.groppi@unipr.it
Haskovec Jan	University of Wien, Austria	jan.haskovec@univie.ac.at
Illner Reinhard	University of Victoria, Canada	rillner@math.uvic.ca
Juntasaro Ekachai	University of Bangkok, Thailand	junta@sut.ac.th
Juntasaro Varangrat	University of Bangkok, Thailand	fengvrj@ku.ac.th
Lafitte Pauline	Universite' de Lille, France	Pauline.Lafitte-Godillon@math.univ-lille1.fr
Lampis Maria	Politecnico di Milano, Italia	marlam@mate.polimi.it
Lisi Meri	Universita' di Siena, Italia	lisi7@unisi.it
Lorenzani Silvia	Politecnico di Milano, Italia	silvia@mate.polimi.it
Mancini Simona	Universite' de Orleans, France	simona.mancini@univ-orleans.fr
Manzini Chiara	Universita' di Firenze, Italia	chiara.manzini@unifi.it
Matthes Daniel	University of Mainz, Germany	matthes@mathematik.uni-mainz.de
Milisic Josipa Pina	University of Mainz, Germany	milisic@mathematik.uni-mainz.de
Mola Gianluca	Politecnico di Milano, Italia	mola@mate.polimi.it
Monaco Roberto	Politecnico di Torino, Italia	roberto.monaco@polito.it
Morandi Cecchi Maria	Universita' di Padova, Italia	mcecchi@math.unipd.it
Mouhot Clement	Universite' de Paris IX, France	cmouhot@ceremade.dauphine.fr
Negulescu Claudia	Universite' Paul Sabatier, Toulouse, France	negulesc@mip.ups-tlse.fr
Nouri Anne	Universite' de Aix-Marseille I, France	Anne.Nouri@cmi.univ-mrs.fr
Pandolfi Bianchi Miriam	Politecnico di Torino, Italia	miriam.pandolfi@polito.it
Petterson Rolf	Chalmers University of Technology, Goteborg, Sweden	rolfp@math.chalmers.se
Pieraccini Sandra	Politecnico di Torino, Italia	sandra.pieraccini@polito.it
Potapenko Irina	Keldysh Institute, Moscow, Russia	firena@yandex.ru
Precioso Juliana Conceicao	Universitat Autonoma Barcelona, Espana	jcprecioso@gmail.com
Puppo Gabriella	Politecnico di Torino, Italia	gabriella.puppo@polito.it
Rosado Linares Jesus	Universitat Autonoma Barcelona, Espana	jrosado@mat.uab.es
Sacchetti Andrea	Universita' di Modena, italia	sacchet@unimo.it
Salvarani Francesco	Universita' di Pavia, Italia	francesco.salvarani@unipv.it
Semplice Matteo	Universita' di Milano, Italia	semplice@mat.unimi.it
Servente Giorgia	Politecnico di Torino, Italia	giorgiaservente@yahoo.com
Soares Ana Jacinta	Universidade do Minho, Braga, Portugal	ajsoares@math.uminho.pt
Spiga Giampiero	Universita' di Parma, Italia	giampiero.spiga@unipr.it
Terraneo Elide	Universita' di Milano, Italia	terraneo@mat.unimi.it
Toscani Giuseppe	Universita' di Pavia, Italia	giuseppe.toscani@unipv.it
Totaro Silvia	Universita' di Siena, Italia	totaro@unisi.it
Trazzi Stefano	Universita' di Ferrara, Italia	trazste@libero.it
Villani Cedric	Ecole Normale Superieure Lyon, France	cedric.villani@umpa.ens-lyon.fr
Vinerean Mirela	University of Karlstad, Sweden	mirela.vinerean@kau.se
Yoshida Hiroaki	Kyoto University, Japan	hiroaki-yos@dsaa.mbox.media.kyoto-u.ac.jp
Zechineli Fernandes Flavia	Universitat Autonoma Barcelona, Espana	zechinel@gmail.com