Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation

The semiconductor Boltzmann-Dirac-Benney equation

$ \partial_t f + \nabla\epsilon(p)\cdot\nabla_x f - \nabla \rho_f(x,t)\cdot\nabla_p f = \frac{\mathcal F_\lambda(p)-f}\tau, \quad x\in\mathbb{R}^d,\ p\in B, \ t>0 $

is a model for ultracold atoms trapped in an optical lattice. The global existence of a solution is shown for small $ \tau>0 $ assuming that the initial data are analytic and sufficiently close to the Fermi-Dirac distribution $ \mathcal F_\lambda $. This system contains an interaction potential $ \rho_f: = \int_B fdp $ being significantly more singular than the Coulomb potential, which causes major structural difficulties in the analysis.

The key technique is based of the ideas of Mouhot and Villani by using Gevrey-type norms which vary over time. The global existence result for small initial data is also generalized to

$ \partial_t f + Lf = Q(f), $

where $ L $ is a generator of an $ C^0 $-group with $ \|e^{tL}\|\leq Ce^{\omega t} $ for all $ t\in\mathbb R $ and $ \omega>0 $ and, where further additional analytic properties of $ L $ and $ Q $ are assumed.