Abstract
This paper considers the linear space-inhomogeneous Boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force. The paper gives results on strong convergence to equilibrium, whent→∞, for general initial data; first in the cutoff case, and then for infinite-range collision forces. The proofs are based on the properties of translation continuity and weak convergence to equilibrium. To handle these problems generalH-theorems (concerning monotonicity in time of convex entropy functionals) are presented. Furthermore, the paper gives general results on collision invariants, i.e., on functions satisfying detailed balance relations in a binary collision.
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References
L. Arkeryd, On the Boltzmann equation,Arch. Rat. Mech. Anal. 45:1–34 (1972).
L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,Arch. Rat. Meck Anal. 77:11–21 (1981).
L. Arkeryd, On the long time behaviour of the Boltzmann equation in a periodic box, Technical Report 23, Department of Mathematics, University of Göteborg (1988).
L. Arkeryd and C. Cercignani, On a functional equation arising in the kinetic theory of gases,Rend. Mat. Acc. Lincei 9:139–149 (1990).
L. Boltzmann,Vorlesungen über Gastheorie, I (Verlag von Johann Ambrosius Barth, Leipzig, 1896).
N. Bellomo, A. Palczewski, and G. Toscani,Mathematical Topics in Nonlinear Kinetic Theory (World Scientific, Singapore, 1989).
T. Carleman,Problémes mathématiques dans la théorie cinétique des gaz (Almqvist-Wiksell, Uppsala, 1957).
C. Cercignani,The Boltzmann Equation and Its Applications (Springer-Verlag, Berlin, 1988).
F. Chvála, T. Gustafsson, and R. Pettersson, On solutions to the linear Boltzmann equation with external electromagnetic force,SIAM J. Math. Anal. 24:583–602 (1993).
F. Chvála and R. Pettersson, Weak solutions of Boltzmann equation with very soft interactions, Preprint, Department of Mathematics, Chalmers University of Technology, 1992-42.
L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations,Arch. Rat. Mech. Anal. 110:73–91 (1990).
T. Dlotko and A. Lasota, On the Tjon-Wu representation of the Boltzmann equation,Ann. Polon. Math. 42:73–82 (1983).
N. Dunford and J. T. Schwartz,Linear Operators I (Interscience, New York, 1958).
T. Elmroth, On theH-function and convergence towards equilibrium for a space-homogeneous molecular density,SIAM J. Appl. Math. 44:150–159 (1984).
W. Greenberg, C. van der Mee, and V. Protopopescue,Boundary Value Problems in Abstract Kinetic Theory (Birkhäuser-Verlag, 1987).
T. Gustafsson, GlobalL p-properties for the spacially homogeneous Boltzmann equation,Arch. Rat. Mech. Anal. 103:1–38 (1988).
R. Illner and H. Neunzert, Relative entropy maximization and directed diffusion equations, Preprint, Department of Mathematics, University of Victoria (1990).
H. G. Kaper, C. G. Lekkerkerker, and J. Hejtmanek,Spectral Methods in Linear Transport Theory (Birkhäuser-Verlag, 1982).
A. Lasota and M. Mackey,Probabilistic Properties of Deterministic Systems (Cambridge University Press, Cambridge, 1988).
K. Loskot and R. Rudnicki, Relative entropy and stability of stochastic semigroups,Ann. Polon. Math. 53:139–145 (1991).
R. Petterson, Existence theorems for the linear, space-inhomogeneous transport equation,IMA J. Appl. Math. 30:81–105 (1983).
R. Pettersson, On solutions and higher moments for the linear Boltzmann equation with infinite-range forces,IMA J. Appl. Math. 38:151–166 (1987).
R. Pettersson, On solutions to the linear Boltzmann equation with general boundary conditions and infinite range forces,J. Stat. Phys. 59:403–440 (1990).
R. Pettersson, On the linear Boltzmann equation with sources, external forces, boundary conditions and infinite range collisions,Math. Mod. Meth. Appl. Sci. 1:259–291 (1991).
R. J. di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equations, global existence and weak stability,Ann. Math. 130:321–366 (1989).
R. J. di Perna and P. L. Lions, Global solutions of Boltzmann equation and the entropy inequality,Arch. Rat. Mech. Anal. 114:47–59 (1991).
M. Reed, and B. Simon,Methods of Modern Mathematical Physics III, Scattering Theory (Academic Press, New York, 1979).
C. Truesdell and R. G. Muncaster,Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York, 1980).
J. Voigt, Functional analytic treatment of the initial boundary value problem for collisionless gases, Habilitations-Schrift, Universität München (1980).
B. Wennberg, Stability and exponential convergence inL p for the spatially homogeneous Boltzmann equation,Nonlinear Analysis, Meth. Appl. 20:935–964 (1993).
B. Wennberg, On an entropy dissipation inequality for the Boltzmann equation.C. R. Acad. Sci. Paris,315 (Série I):1441–1446 (1992).
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Pettersson, R. On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation. J Stat Phys 72, 355–380 (1993). https://doi.org/10.1007/BF01048054
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DOI: https://doi.org/10.1007/BF01048054