Abstract
An existence theorem is proved for homoenergetic affine flows described by the Boltzmann equation. The result complements the analysis of Truesdell and of Galkin on the moment equations for a gas of Maxwellian molecules. Existence of the distribution function is established here for a large class of molecular models (hard sphere and angular cut-off interactions). Some of the data lead to an implosion and infinite density in a finite time, in agreement with the physical picture of the associated flows; for the remaining set of data, global existence is shown to hold.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Truesdell, “On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, II”, Journal of Rational Mechanics and Analysis 5, 55–128 (1956).
В. С. галкин (V. S. galkin), ≪Об одном решений кинетического уравнения≫, Прикладная Математика и Механика 20, 445–446 (1956).
V. S. Galkin, “On a class of solutions of Grad's moment equations”, PMM 22, 532–536 (1958).
V. S. Galkin, “One-dimensional unsteady solutions of the equations for the kinetic moments of a monatomic gas”, PMM 28, 226–229 (1964).
V. S. Galkin, “Exact solutions of the kinetic moment equations of a mixture of monatomic gases”, Fluid Dynamics 1, 29–34 (1966).
C. Truesdell & R. S. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, New York, Academic Press (1980).
A. A. Nikol'skii, “The simplest exact solutions of the Boltzmann equation for the motion of a rarefied gas”, Soviet Physics-Doklady 8, 633–635 (1964).
A. A. Nikol'skii, “The three-dimensional expansion-contraction of rarefied gas with power interaction functions”, Soviet Physics-Doklady 8, 639–641 (1964).
M. Krook & T. T. Wu, “Formation of Maxwellian tails”, Physical Review Letters 36, 1107–1109 (1976).
M. Krook & T. T. Wu, “Exact solutions of the Boltzmann equation”, The Physics of Fluids 20, 1589–1595 (1977).
R. G. Muncaster, “On generating exact solutions of the Maxwell-Boltzmann equation”, Archive for Rational Mechanics and Analysis 70, 79–90 (1979).
C. Cercignani, Mathematical Methods in Kinetic Theory, New York, Plenum Press, and London, McMillan (1969).
C. Cercignani, The Boltzmann Equation and its Applications, New York, Springer Verlag (1987).
L. Arkeryd, “On the Boltzmann equation. Part I: Existence”, Archive for Rational Mechanics and Analysis 45, 1–16 (1972).
L. Arkeryd, “On the Boltzmann equation. Part II: The full initial value problem”, Archive for Rational Mechanics and Analysis 45, 17–34 (1972).
A. Ya. Povzner, “The Boltzmann equation in the kinetic theory of gases”, American Mathematical Society Translations (2) 47, 193–216 (1962).
Author information
Authors and Affiliations
Additional information
Communicated by R. G. Muncaster
Rights and permissions
About this article
Cite this article
Cercignani, C. Existence of homoenergetic affine flows for the Boltzmann equation. Arch. Rational Mech. Anal. 105, 377–387 (1989). https://doi.org/10.1007/BF00281497
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00281497