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

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. 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.