On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: Averaging compactness of weak solutions

The paper considers macroscopic behavior of a Fermi - Dirac particle system. We prove the L-1-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi - Dirac particles in a periodic box with the collision kernel b(cos theta)| v - v*|(gamma), which corresponds to very soft potentials: - 5 < gamma <= - 3 with a weak angular cutoff: integral(pi)(0) b(cos theta) sin(3) theta d theta < infinity. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff - Young inequality, the L-infinity-bounds of the solutions, and a specific property of the value-range of the exponent gamma. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.