# American Institute of Mathematical Sciences

April  2019, 12(2): 445-482. doi: 10.3934/krm.2019019

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

 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Received  June 2018 Published  November 2018

Fund Project: The author was partially funded by the Austrian Science Fund (FWF) project F 65

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.
Citation: Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019
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