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- W2768430149 abstract "A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice: [ partial_t f + nabla_pepsilon(p)cdotnabla_x f - nabla_x n_fcdotnabla_p f = n_f(1- n_f)(mathcal{F}_f-f), quad xinmathbb{R}^d, pinmathbb{T}^d, t>0. ] This system contains an interaction potential $n_f(x,t):=int_{mathbb{T}^d}f(x,p,t)dp$ 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, $epsilon(p) = -sum_{i=1}^d$ $cos(2pi p_i)$ is the dispersion relation and $mathcal{F}_f$ 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." @default.
- W2768430149 created "2017-12-04" @default.
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- W2768430149 date "2017-11-16" @default.
- W2768430149 modified "2023-09-27" @default.
- W2768430149 title "Semiconductor Boltzmann-Dirac-Benney equation with BGK-type collision operator: existence of solutions vs. ill-posedness" @default.
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