Banasiak, Jacek; Barletti, Luigi On the existence of propagators in stationary Wigner equation without velocity cut-off. (English) Zbl 0990.82019 Transp. Theory Stat. Phys. 30, No. 7, 659-672 (2001). L. Barletti and P. F. Zweifel have applied [Transp. Theor. Stat. Phys. 30, 507-520 (2001; Zbl 1006.82032)] a parity decomposition method to recast the one-dimensional, stationary Wigner equation with inflow boundary conditions into two decoupled evolution equations but with coupling remaining in the initial conditions. The singularity introduced by the division of the velocity \(\nu\) forced the authors to perform the analysis in a simplified situation with the velocity cut-off close to zero. In this note we shall show that the operators introduced in the paper mentioned above, generate evolution families in suitably weighted \(L^2\) spaces, without introducing the velocity cut-off. These spaces are different for each equation, though there is a common space in which both evolutions take place provided the initial conditions are appropriately selected. This allows to solve the two evolution equations but falls short of providing the solution of the original problem with complete inflow boundary conditions. Cited in 1 Document MSC: 82C40 Kinetic theory of gases in time-dependent statistical mechanics 35Q35 PDEs in connection with fluid mechanics 47D06 One-parameter semigroups and linear evolution equations Keywords:stationary Wigner equation; parity decomposition method; evolution operators Citations:Zbl 1006.82032 PDFBibTeX XMLCite \textit{J. Banasiak} and \textit{L. Barletti}, Transp. Theory Stat. Phys. 30, No. 7, 659--672 (2001; Zbl 0990.82019) Full Text: DOI