×

On the existence of propagators in stationary Wigner equation without velocity cut-off. (English) Zbl 0990.82019

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.

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

Citations:

Zbl 1006.82032
PDFBibTeX XMLCite
Full Text: DOI