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: $$\partial_t f + \nabla_p\epsilon(p)\cdot\nabla_x f - \nabla_x n_f\cdot\nabla_p f = n_f(1- n_f)(\mathcal{F}_f-f), \quad x\in\mathbb{R}^d, p\in\mathbb{T}^d, t>0.$$ 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$.=0$) - 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.