# zbMATH — the first resource for mathematics

Are there more than five linearly-independent collision invariants for the Boltzmann equation? (English) Zbl 0718.60113
Summary: The problem of finding the summational collision invariants for the Boltzmann equation is tackled with the aim of proving that the most general solution of the problem is not different from the standard one even when the equation defining a collision invariant $$\psi$$ is only satisfied almost everywhere in $$R^3\times R^3\times S^2$$. The collision invariant $$\psi$$ is assumed to be in the Hilbert space $$H_{\omega}$$ of the functions which are square integrable with respect to a Maxwellian weight.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C40 Kinetic theory of gases in time-dependent statistical mechanics
##### Keywords:
kinetic theory; Boltzmann equation; collision invariant
Full Text:
##### References:
 [1] C. Cercignani,Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1969). · Zbl 0191.25103 [2] C. Cercignani,The Boltzmann Equation and Its Applications (Springer-Verlag, New York, 1988). · Zbl 0646.76001 [3] L. Boltzmann, Über das Wärmegleichgewicht von Gasen, auf welche äussere Kräfte wirken,Sitzungsber. Akad. Wiss. Wien 72:427-457 (1875). · JFM 07.0683.03 [4] L. Boltzmann, Über die Aufstellung und Integration der Gleichungen, welche die Molekular bewegungen in Gasen bestmmen,Sitzungsber. Akad. Wiss. Wien 74:503-552 (1876). [5] T. H. Gronwall, A functional equation in the kinetic theory of gases,Ann. Math. (2) 17:1-4 (1915). · JFM 45.0514.01 · doi:10.2307/2007210 [6] T. H. Gronwall, Sur une équation fonctionelle dans la théorie cinétique des gaz,C. R. Acad. Sci. Paris 162:415-418 (1916). [7] T. Carleman,Problèmes Mathématiques dans la Théorie Cinétique des Gaz (Almqvist & Wiksell, Uppsala, 1957). · Zbl 0077.23401 [8] H. Grad, On the kinetic theory of rarified gases,Commun. Pure Appl. Math. 2:331-407 (1949). · Zbl 0037.13104 · doi:10.1002/cpa.3160020403 [9] C. Truesdell and R. G. Muncaster,Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York, 1980). [10] L. Arkeryd, On the Boltzmann equation. Part II: The full initial value problem,Arch. Rat. Mech. Anal. 45:17-34 (1972). · Zbl 0245.76060
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.