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

The global existence of a solution of 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 shown for small $\tau>0$ assuming that the initial data are analytic and sufficiently close to $\mathcal F_\lambda$. This system contains an interaction potential $\rho_f(x,t):=\int_{B}f(x,p,t)dp$ being significantly more singular than the Coulomb potential, which causes major structural difficulties in the analysis. The semiconductor Boltzmann-Dirac-Benney equation is a model for ultracold atoms trapped in an optical lattice. Hence, the dispersion relation is given by $\epsilon(p) = -\sum_{i=1}^d$ $\cos(2\pi p_i)$, $p\in B=\mathbb{T}^d$ due to the optical lattice and the Fermi-Dirac distribution $\mathcal F_\lambda(p)=1/(1+\exp(-\lambda_0-\lambda_1\epsilon(p)))$ describes the equilibrium of ultracold fermionic clouds. This equation is closely related to the Vlasov-Dirac-Benney equation with $\epsilon(p)=\frac{p^2}2$, $p\in B=\mathbb R^d$ and r.h.s$.=0$, where the existence of a global solution is still an open problem. So far, only local existence and ill-posedness results were found for theses systems. 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 shown for a far more general setting, namely $$\partial_t f + Lf=Q(f),$$ 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.