SEMICONDUCTOR BOLTZMANN-DIRAC-BENNEY EQUATION WITH A BGK-TYPE COLLISION OPERATOR: EXISTENCE OF SOLUTIONS VS. ILL-POSEDNESS

摘要

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice: &(p)epsilon(p).del(x)f-del(x)nf.del(p)f=n(f)(1-n(f))(F-f-f), x epsilon R-d, p is an element of T-d, t 0 . This system contains an interaction potential n(f) (x , t) := integral(Td) 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, epsilon(p) = -Sigma(d)(i=1) cos(2 pi p(i)) is the dispersion relation and F-f denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on f in this context.