Lecturer: Christian Schmeiser

Kinetic transport equations are mathematical descriptions of the dynamics of large particle ensembles in terms of a phase space (i.e. position-velocity space) distribution function. They are sometimes called mesoscopic models, which places them between microscopic models, where the dynamics of the individual particles are described, and macroscopic or continuum mechanics models, where the material is described by a finite number of position dependent quantities such as the mass density, the mean velocity, the stress tensor, the temperature etc. Mathematically, kinetic equations typically are integro-differential equations of a particular form. Due to their importance from a modeling point of view, their mathematical theory has become a highly developed subfield of the theory of partial differential equations. The undisputedly most important kinetic equation is the Boltzmann equation for hard spheres, where the underlying particle system is an idealized 3D-billiard. It is well accepted as a model for ideal gases which, on a macroscopic level, are typically described by the Euler or the Navier-Stokes equations. The latter systems can be formally derived from the Boltzmann equation by procedures called macroscopic limits. The main goals of this course are to present

• a formal derivation of the Boltzmann equation from the microscopic model,
• a proof of a variant of the DiPerna-Lions theorem on the existence of large global solutions of the Boltzmann equation, and
• an introduction to a recent result by Golse/Saint-Raymond on the convergence of DiPerna-Lions solutions of the Boltzmann equation to so called Leray solutions of the incompressible Navier-Stokes equations in a combined macroscopic-small-Mach-number limit.

The presented material is mostly taken from books by C. Cercignani, R. Illner, M. Pulvirenti, and by L. Saint-Raymond.

Schedule:
Mon, 12:30-14:00, seminar room, 8th floor; starting on March 6


News:

• The course is cancelled on March 13 (rector's day), March 27, April 24, June 26
• Lecture notes