Semiconductor Boltzmann-Dirac-Benney equation with BGK-type collision operator: existence of solutions vs. ill-posedness

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice: $${\mathrm{\partial }}_{t}f+{\mathrm{\nabla }}_pϵ(p)\cdot {\mathrm{\nabla }}_{x}f-{\mathrm{\nabla }}_x{n}_f\cdot {\mathrm{\nabla }}_{p}f=n_f(1-n_f)({\mathcal{F}}_f-f),x\in {\mathbb{R}}^d,p\in {\mathbb{T}}^d,t>0.$$ This system contains an interaction potential $n_f(x,t):={\int }_{{\mathbb{T}}^d}f(x,p,t)dp$ being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore, $\epsilon(p) = -\sum_{i=1}^d$ $\cos (2\pi p_i)$ is the dispersion relation and ${\mathcal{F}}_f$ denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on $f$ in this context. In a dilute plasma - without collisions (r.h.s$\mathrm{.}=0$) - this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.