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A semiconductor Boltzmann equation with a non-linear BGK-type collisionoperator is analyzed for a cloud of ultracold atoms in an optical lattice:  \[  \partial_t f + \nabla_p\epsilon(p)\cdot\nabla_x f - \nabla_x n_f\cdot\nabla_pf = n_f(1- n_f)(\mathcal{F}_f-f), \quad x\in\mathbb{R}^d, p\in\mathbb{T}^d,t>0.  \]  This system contains an interaction potential$n_f(x,t):=\int_{\mathbb{T}^d}f(x,p,t)dp$ being significantly more singularthan the Coulomb potential, which is used in the Vlasov-Poisson system. Thiscauses 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 dependsnon-linearly on $f$ in this context.  In a dilute plasma - without collisions (r.h.s$.=0$) - this system is closelyrelated to the Vlasov-Dirac-Benney equation. It is shown for analytic initialdata that the semiconductor Boltzmann equation possesses a local, analyticsolution. Here, we exploit the techniques of Mouhout and Villani by usingGevrey-type norms which vary over time. In addition, it is proved that thisequation is locally ill-posed in Sobolev spaces close to some Fermi-Diracequilibrium distribution functions.

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