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

Abstract

The global existence of a solution of the semiconductor Boltzmann-Dirac-Benney equation \[ \partial_t f + \nablaε(p)\cdot\nabla_x f - \nabla ρ_f(x,t)\cdot\nabla_p f = \frac{\mathcal F_λ(p)-f}τ, \quad x\in\mathbb{R}^d,\ p\in B, \ t>0 \] 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 $\Vert e^{tL}\Vert \le C{e}^{\omega t}$ for all $t\in \mathbb{R}$ and $\omega >0$ and, where further additional analytic properties of $L$ and $Q$ are assumed.