×

zbMATH — the first resource for mathematics

Existence theorems for certain kinetic equations and large data. (English) Zbl 0654.76073
The author mainly analyzes the Boltzmann equation in one spatial dimension with a bounded weight function S in the collision operator. For this equation, the global \(L^ 1\)-existence and uniqueness is proved on a bounded x-interval. The theorem is proved when velocities are required to be bounded and the contribution from large velocities is treated as being a small perturbation.
Reviewer: I.Grosu

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L. Arkeryd, On the Boltzmann equation II, Arch. Rational Mech. Anal. 45, 17–34 (1972). · Zbl 0245.76060
[2] L. Arkeryd, An existence theorem for a modified space-inhomogeneous, nonlinear Boltzmann equation, Bull. Am. Math. Soc. 78, 610–614 (1972). · doi:10.1090/S0002-9904-1972-13028-3
[3] L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch. Rational Mech. Anal. 86, 85–97 (1984). · Zbl 0598.76091 · doi:10.1007/BF00280649
[4] L. Arkeryd, On the Boltzmann equation in unbounded space, Comm. Math. Phys. 105, 205–219 (1986). · Zbl 0606.76094 · doi:10.1007/BF01211099
[5] L. Arkeryd, R. Esposito, & M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data, Comm. Math. Phys. 111, 393–407 (1987). · Zbl 0663.76080 · doi:10.1007/BF01238905
[6] T. Beale, Large time behaviour of discrete velocity Boltzmann equations, Comm. Math. Phys. 106, 659–678 (1986). · Zbl 0637.76070 · doi:10.1007/BF01463401
[7] C. Cercignani, Global existence of solutions for a model Boltzmann equation, Jour. Stat. Phys., to appear. · Zbl 0960.82513
[8] C. Cercignani, Small data existence for the Enskog equation inL 1,preprint (1987).
[9] A. Glikson, On the existence of general solutions of the initial value problem for the nonlinear Boltzmann equation, Arch. Rational Mech. Anal. 45, 35–46 (1972). · Zbl 0247.76067 · doi:10.1007/BF00253394
[10] G. Toscani. On the Cauchy problem for the discrete Boltzmann equation with initial values in L 1 + (R), preprint (1987). · Zbl 0635.76078
[11] S. Ukai, On the existence of global solutions of mixed problems for nonlinear Boltzmann equation, Proc. Japan Acad. 50, 179–184 (1974). · Zbl 0312.35061 · doi:10.3792/pja/1195519027
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.