Abstract
We extend the existence theorem recently proved by Hamdache for the initial-boundary-value problem for the nonlinear Boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal. In the latter case a cut-off for large speeds is introduced in the collision term of the Boltzmann equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Hamdache, Initial boundary value problems for Boltzmann equation. Global existence of weak solutions. Arch. Rational Mech. Anal. 119, 309–353 (1992).
R. DiPerna & P. L. Lions, On the Cauchy problem for Boltzmann equations, Ann. of Math. 130, 321–366 (1989).
C. Cercignani, Mathematical Methods in Kinetic Theory, 2nd edition, Plenum Press, New York (1990).
C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York (1987).
L. Arkeryd, C. Cercignani & R. Illner, Measure solutions of the steady Boltzmann equation in a slab, Commun. Math. Phys. 142, 285–296 (1991).
L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations, Arch. Rational Mech. Anal. 110, 73–91 (1990).
C. Cercignani, Equilibrium states and trend to equilibrium in a gas according to the Boltzmann equation, Rend. Mat. Appl. 10, 77–95 (1990).
L. Arkeryd, On the strong L 1 trend to equilibrium for the Boltzmann equation, Studies in Appl. Math. 87, 283–288 (1992).
S. Kawashima, Global solutions to the initial-boundary value problems for the discrete Boltzmann equation, Nonlin. Anal. 17, 577–597 (1991).
J. Darrozés & J.-P. Guiraud, Généralisation formelle du théorème H en présence de parois. Applications, C.R. Acad. Sci. Paris A 262, 1368–1371 (1966).
C. Cercignani & M. Lampis, Kinetic models for gas-surface interactions, Transp. Th. Stat. Phys. 1, 101–114 (1971).
C. Cercignani, Scattering kernels for gas-surface interactions, Transp. Th. Stat. Phys. 2, 27–53 (1972).
S. Ukai, Solutions of the Boltzmann equation, in Pattern and Waves — Qualitative Analysis of Nonlinear Differential Equations, 37–96 (1986).
C. Cercignani, On the initial-boundary value problem for the Boltzmann equation, Arch. Rational Mech. Anal. 116, 307–315 (1992).
J. Voigt, Functional analytic treatment of the initial boundary value problem for collisionless gases, Habilitationsschrift, Univ. München (1980).
R. Beals & V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl. 121, 370–405 (1987).
M. Cannone & C. Cercignani, A trace theorem in kinetic theory, Appl. Math. Lett. 4, 63–67 (1991).
F. Golse, P. L. Lions, B. Perthame & R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76, 110–125 (1988).
L. Arkeryd & C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann Equation, Comm. Partial Diff. Eqs. 14, 1071–1089 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arkeryd, L., Cercignani, C. A global existence theorem for the initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rational Mech. Anal. 125, 271–287 (1993). https://doi.org/10.1007/BF00383222
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00383222