2012 Young Researchers Workshop:
Kinetic Description of Multiscale Phenomena

Computations on the spectral gap for the linearized Boltzmann operator

Chenglong Zhang

University of Texas at Austin

The linearized Boltzmann Collision Operator contains important information on the existence and (exponential) decay to equilibrium of the solutions to the original nonlinear Boltzmann equations. This is also the linearized version of the well-known Cercignani's conjecture on the estimates of the entropy production functional. The key part is to answer whether there is a spectral gap for the linear operator and what it looks like. As will be discussed, the behavior of the spectral gaps largely depends on the types of the collision kernels--both the intermolecular potentials and angular cross-sections. This talk will survey different analytical and numerical treatments for both integrable and non-integrable angular cross-sections. For the integrable ones, some simple DG discretizations can be applied; for non-integrable ones, we work in Fourier space through a Plancherel-like identity and show how the singularity of the angular part is absorbed in the weights. The spectral gap problem is finally reduced to minimizing the Rayleigh quotients for a symmetric semi positive-definite real matrix. Some numerical results will be given.