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

Abstract

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice: \begin{document} $ \partial _t f + \nabla _pe(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. $ \end{document} This system contains an interaction potential \begin{document}$n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$\end{document} 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, \begin{document}$e(p) = -\sum_{i = 1}^d$\end{document} \begin{document}$\cos(2π p_i)$\end{document} is the dispersion relation and \begin{document}$\mathcal{F}_f$\end{document} denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on \begin{document}$f$\end{document} in this context. In a dilute plasma—without collisions (r.h.s \begin{document}$. = 0$\end{document} )—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.