Isaac Newton Institute for Mathematical Sciences

Partial Differential Equations in Kinetic Theories

16 August - 22 December 2010

Organisers: Professor JA Carrillo (Barcelona), Professor S Jin (Wisconsin) and Professor PA Markowich (Cambridge)

Scientific Advisors: Professor Y Guo (Brown), Professor R Illner (Victoria), Professor B Perthame (Paris), Professor A Stuart (Warwick), Professor JF Toland (Bath) and Professor G Toscani (Pavia)

Programme Theme

Kinetic equations occur naturally in the modelling of the collective motion of large individual particle ensembles such as molecules in rarefied gases, beads in granular materials, charged particles in semiconductors and plasmas, dust in the atmosphere, cells in biology, or the behaviour of individuals in economical trading …  Generally, huge interacting particle systems cannot efficiently be described by the individual dynamics of all particles due to overwhelming complexity but clearly some input from the microscopic behaviour is needed in order to bridge from microscopic dynamics to the macroscopic world, typically rendered in terms of averaged quantities. This leads to classical equations of mathematical physics: the Boltzmann equation of rarified gas dynamics, the fermionic and bosonic Boltzmann equations and the relativistic Vlasov-Maxwell system of particle physics, the quantistic Wigner-Poisson system, to name just a few.

Kinetic theory has produced as a spin-off many new mathematical tools in the last 20 years: renormalized solutions of transport equations by R DiPerna and P-L Lions, averaging lemmas by the French kinetic school, entropy dissipation tools which have been extended methodologically and used far beyond kinetic theory are just some highlights of new analytical PDE methods stemming from kinetic theory. Another recent landmark in this field has been the proof of the hydrodynamic limit process of the renormalized solutions of the Boltzmann equation towards (weak) Leray solutions of the Navier-Stokes equations by F Golse and L Saint-Raymond. On the other hand, kinetic theory has different scientific viewpoints ranging from applied mathematical and physical modelling to stochastic analysis, numerical analysis of PDEs and in many important cases extensive numerical simulations.

The main objective of this program is aimed at advancing Partial Differential Equations (PDEs) research in kinetic theories and its impact in the applied sciences highlighting selected modern application areas. This effort has to be understood from a global perspective of research in PDEs bringing together mathematical modelling, analysis, numerical schemes and simulation in a feedback loop of synergies. The three selected newly emerging application areas of kinetic theories are kinetic modelling in biology, coupled fluid-particle models and PDE Models for quantum fluids.

As part of the program, several workshops will take place in the Newton Institute, and one satellite workshop at the ICMS in Edinburgh. There are also two series of seminars which will take place weekly at the Newton Institute: