×

Measure solutions for the steady linear Boltzmann equation in a slab. (English) Zbl 0940.35165

The authors of this interesting paper consider a one-dimensional linear stationary Boltzmann equation in a slab, \(v_x\partial_xf= (Qf)(x,{\mathbf v})\) (\(x\in [0,L]\), \({\mathbf v}\in \mathbb{R}^3\)). \(Qf\) is so-called collision term with some collision frequency. The solution \(f\) of the linear Boltzmann equation is presented by \(f(x,{\mathbf v})=\widetilde Af(x,{\mathbf v})\), that is, written as a fixed point problem. It turns out that \(\widetilde Af\) is linear but not homogeneous. It is proved via Tychonoff’s fixed-point theorem the existence of a steady state solution of the considered problem in the space of measures.

MSC:

35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Greenberg W., Operator Theory: Advances and Applications 23 (1987)
[2] Halmos P. R., Measure theory (1950)
[3] DOI: 10.1063/1.1703916 · Zbl 0132.44902 · doi:10.1063/1.1703916
[4] DOI: 10.1016/0022-247X(71)90238-1 · Zbl 0214.37105 · doi:10.1016/0022-247X(71)90238-1
[5] Suhadolc A., Mathematica Balkanica 3 pp 514– (1973)
[6] DOI: 10.1007/BF02413476 · Zbl 0347.45002 · doi:10.1007/BF02413476
[7] DOI: 10.1007/978-1-4612-1039-9 · doi:10.1007/978-1-4612-1039-9
[8] Dunford N., Linear Operators I, II, III (1958)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.