Boltzmann Models in Kinetic Theory (November 711, 2011)
 Kazuo Aoki
(Kyoto University)  Yan Guo
(Brown University)  Shi Jin
(University of Wisconsin)  Lorenzo Pareschi
(University of Ferrara)  Laure SaintRaymond
(Universite Paris VI)
The celebrated Boltzmann equation is the foundation of the kinetic theory for dilute collections of particles, which undergo elastic binary collisions. The Boltzmann theory is at the center of a series of multiscaled physical models that connect microscopic multiparticle models to macroscopic fluid models such as the NavierStokes equations:
The first arrow refers the BoltzmannGrad limit, while the second arrow refers to various hydrodynamic limits which lead to the fundamental equations of fluids. The Boltzmann theory therefore provides a practical tool and machinery for deriving macroscopic models in broad physical applications. Due to its importance, there has been an explosion of mathematical studies, both theoretical and numerical, for the Boltzmann equation. A major open problem that remains is to determine whether or not smooth initial data would lead to a unique globalintime solution of the Boltzmann equation. Nevertheless, there have been exciting new developments in recent years. The focus of the program is to bring computational and theoretical people together to investigate problems of fundamental importance.
Problem 1: Boundary Effects.
Boundary effects play an important role in the dynamics of particles confined in a bounded region. Yet its mathematical study is at an early stage. This is due to the fact that solutions to the Boltzmann equation in general will develop singularities. The focus is to investigate the formation and propagation of singularities, both from numerical and theoretical points of view.
Problem 2: Hydrodynamic Limits.
There have been lots of studies of hydrodynamic limits of the Boltzmann equation. The focus in our program will be on error estimates and higherorder expansions of hydrodynamic limits both from the theoretical point of view and from the point of view of numerical simulation. Boundary and initial layer analysis for hydrodynamic limits, which has been barely studied, is an important area that is ready for investigation.
Problem 3: BoltzmannGrad Limit.
There has been little mathematical work in this direction since the work of Lanford. The focus will be on boundary effects in the BoltzmannGrad limit, and on the application of Lanford?s proof to establish the BoltzmannGrad limit for other particle systems of physical importance. Numerical simulations at the particle level will play an key role.



Monday  October 31st 2011  

Time  Description  Speaker  Location  Abstracts 
1:30  2:00  Coffee/Tea Break  11th Floor Collaborative Space  
2:00  4:00  An introduction to Collisional (Boltzmanntype) models in Kinetic theory  Robert Strain, University of Pennsylvania  11th Floor Lecture Hall  
Tuesday  November 1st 2011  
Time  Description  Speaker  Location  Abstracts 
2:30  3:00  Coffee/Tea Break  11th Floor Collaborative Space  
Wednesday  November 2nd 2011  
Time  Description  Speaker  Location  Abstracts 
2:30  3:00  Coffee/Tea Break  11th Floor Collaborative Space  
Thursday  November 3rd 2011  
Time  Description  Speaker  Location  Abstracts 
1:50  2:00  ICERM Long Term Visitor Group Photo  11th Floor Lecture Hall  
2:00  4:00  An introduction to spectral approximaton for Boltzmann equation  Francis Filbet, Universite Claude Bernard, Lyon I  11th Floor Lecture Hall  
4:00  4:30  Coffee/Tea Break  11th Floor Collaborative Space  
Friday  November 4th 2011  
Time  Description  Speaker  Location  Abstracts 
10:00  12:00  An introduction to spectral approximaton for Boltzmann equation  Francis Filbet, Universite Claude Bernard, Lyon I  10th Floor Classroom  
2:00  4:00  Asymptotic analysis for boundaryvalue problems of the Boltzmann equation  Kazuo Aoki, Kyoto University  10th Floor Classroom  
4:00  4:30  Coffee/Tea Break  10th Floor 