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- Title
Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness.
- Authors
Braukhoff, Marcel
- 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: ∂tf + ∇pε(p) ⋅ ∇xf − ∇xnf ⋅ ∇pf = nf(1 − nf)(Ff − f), x ∈ Rd,p ∈ Td,t > 0. This system contains an interaction potential nf(x,t) := ∈ tTd 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) = −∑di=1 cos(2πpi) is the dispersion relation and Ff 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.
- Subjects
DIRAC equation; SOBOLEV spaces; OPTICAL lattices; SEMICONDUCTORS; ELECTRONS
- Publication
Kinetic & Related Models, 2019, Vol 12, Issue 2, p445
- ISSN
1937-5093
- Publication type
Article
- DOI
10.3934/krm.2019019