In the study of semiconductors, adequate simplifying models are of big interest.
A fermionic gas in an optical lattice is a physical model for a semiconductor,
which can mathematically be describe by a semi-classical Boltzmann equation with
some singular potential. Facilitating the collision term, a non-linear relaxation
time approximation of the collision operator will be presented during the talk.
Neglecting all collisions, the Boltzmann equation reduces to a Vlasov equation.
For this equation, we discuss its ill-posedness in the Hadamard sense and
well-posedness in some analytic function spaces.