Semiconductor Boltzmann-Dirac-Benney equation with a 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 _pε(p)·\nabla _x f - \nabla _x n_f·\nabla _p f = n_f(1- n_f)(\mathcal{F}_f-f),\;\;\;\; x∈\mathbb{R}^d, p∈\mathbb{T}^d, t>0. $

This system contains an interaction potential $n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$ 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, $ε(p) = -\sum_{i = 1}^d$ $\cos(2π p_i)$ 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.