Abstract
In this chapter we shall describe the mathematically more advanced part of the theory of the Boltzmann equation, based on the use of techniques of functional analysis. The main results refer to existence and uniqueness theorems and the qualitative behaviour of the solutions of the Boltzmann equation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
X Carleman, “Acta Math.”, 60, 91 (1933).
O. Lanford, III, in “Proceedings of the 1974 Batteile Rencontre on Dynamical Systems”, J. Moser, Ed., Lecture Notes in Physics, 35, 1, Springer, Berlin (1975).
T. Carleman, “Problèmes Mathématiques dans la Theorie Cinétique ses Gaz”, Almqvist and Wiksells, Uppsala (1957).
H. B. Maslova and R. P. Tchubenko, “Dokl. Akad. Nauk SSSR”, 202, no. 4, 800(1972).
H. B. Maslova and R. P. Tchubenko, “Vestnik Leningrad Univ., no. 7, 109(1976).
P. Gluck, “Transport Theory Statist Phys”, 9, 43 (1980).
E. Wild, “Proc, Camb. Philos. Soc”, 47, 602 (1951).
D. Morgenstern, “Proc. Nat. Acad. Sci. U.S.A.”, 40, 719 (1954).
L. Arkeryd, “Arch. Rational Mech. Anal.”, 45, 1 (1972).
L. Arkeryd, “Arch. Rational Mech. Anal.”, 45, 17 (1972).
B. Levï, “Rend. Istit. Lombardo Sci. Lett. (2)”, 39, 775 (1906).
A. YA. Povzner, “Mat. Sbornik”, 58, 65 (1962).
G. Truesdell, “J. Rational Mech. Anal”, 5, 55 (1956).
R. E. Edwards, “Functional Analysis”, Holt, Rinehart and Winston, New York (1965).
L. Arkeryd, “Arch. Rational Mech. Anal”, 77, 11 (1981).
T. Elmroth, “Arch. Rational Mech. Anal.”, 2, 1 (1983).
T. Elmroth, in “Kinetic Theories and the Boltzmann Equation”, G. Cercignani, Ed, Lecture Notes in Mathematics, 1048, 192, Springer, Berlin (1984).
T. Elmroth, “SIAM J. Appl. Math.”, 44, 150 (1984).
T. Gustafsson, “Arch. Rational Mech. Anal.”, 92, 23 (1986).
G. Truesdell and R. G. Muncaster, “Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas”, Academic Press, New York (1980).
G. Di Blasio, “Comm. Math. Phys.”, 38, 331 (1974).
V. BoFFi and G. Spiga, “J. Math. Phys.”, 23, 1859 (1982).
C. Cercignani, “Arch. Mech.”, 34, 231 (1982).
D. Morgenstern, “J, Rational Mech. Anal.”, 4, 533 (1955).
C. Cercignani, “Transport Theory Statist. Phys.” (to appear, 1987).
L. Arkeryd, “Transport Theory Statist. Phys.”, 15, 673 (1986).
C. Cercignani, W. Greenberg and P. Zweifel, “J. Stat. Phys.”, 20, 449(1979).
H. Spohn, “J. Stat. Phys.”, 20, 463 (1979).
W. Greenberg, J. Voigt and P. Zweifel, “J. Stat. Phys”, 21, 649 (1979).
R. Gatignol, “Theorie Cinétique des Gaz à Répartition Discrète de Vitesses”, Lectures Notes in Physics, 36, Springer, Berlin (1975).
J. E. Broadwell, “Phys. Fluids”, 7, 1243 (1964).
M. Shinbrot, in “Mathematical Problems in the Kinetic Theory of Gases”, D. G Pack and H. Neunzert, Eds., Lang, Frankfurt (1980).
T. Nishida and M. Mimura, “Proc. Japan. Acad.”, 50, 812 (1974).
L. Tartar, “Séminaire Goulaouic-Schwartz”, Ecole Polytechnique, Paris (1975).
H. Cabannes, “J. Méc. Théor. Appl.”, 17, 1 (1978).
H. Cabannes, “C.R. Acad. Sci, Paris”, série A, 284, 269 (1977).
H. Cabannes, “Mech. Res. Comm.”, 10, 317 (1983).
R. Illner, “Habilitationsschrift,” Universität Kaiserslautern, (1981).
Y. Shizuta and S. Kawashima, “Hokkaido Math. J.”, 14, 249 (1985).
C. Cercignani, “C.R. Acad. Sci. Paris”, série I, 301, 89 (1985).
C. Cercignani, and M Shinbrot, “Comm. Partial Differential Equations” (to appear, 1987).
T. Platkowski and R. Illner, “Siam Rev.” (to appear, 1987).
U. Frisch, B. Hasslacher and Y. Pomeau, “Phys. Rev. Lett.”, 56, 1505(1986).
L. Arkeryd, “Arch. Rational Mech. Anal.”, 86, 85 (1984).
S. Albeverio, J. E. Fenstad, R. Hooegh-Krohn and T. Lindstrom, “Nonstandard Methods in Stochastic Analysis and Mathematical Physics”, Academic Press, New York (1986).
K. D. Stroyan and W. A. J. Luxenburg, “Introduction to the Theory of Infinitesimals”, Academic Press, New York (1976).
A. E. Hurd and P. Loeb, “An Introduction to Nonstandard Real Analysis”, Academic Press, New York (1985).
A. Robinson, “Non-standard Analysis”, North-Holland, Amsterdam (1966).
P. Loeb, “Trans. Math. Soc, 211, 113 (1975).
T. Elmroth, Chalmers University of Technology, Preprint 1984-23 (1984).
H. Grad, in “Handbuch der Physik”, S. Flügge, Ed., Vol. Xii, Sect. 26, Springer, Berlin (1958).
A. Glickson, “Arch. Rational Mech. Anal.”, 45, 5 and 47, 389 (1972).
S. Kaniel and M. Shinbrot, “Comm. Math. Phys.”, 58, 65 (1978).
M. Shinbrot, “Math. Methods Appl. Sci.”, 6, 539 (1984).
F. G. King, Ph.D. Dissertation, Dept. of Mathematics, Univ. of California at Berkeley (1975).
A. Hurd, Preprint (1986).
H. Spohn, in “Kinetic Theories and the Boltzmann Equation”, C. Cercignani, Ed., Lecture Notes in Mathematics, 1048, 207 Springer, Berlin (1984).
H. Spohn, in “Nonequilibrium Phenomena I. The Boltzmann Equation”, J. L. Lebowitz and E. W. Montroll, Eds., North-Holland, Amsterdam (1983).
C. Cercignani, “Comm. Pure Appl. Math.”, 36, 479 (1983).
H. Grad, in “Rarefied Gas Dynamics”, J. A. Laurmann, Ed., Vol. I, p. 26, Academic Press, New York (1963).
H. Grad, “Siam J.”, 13, 259 (1965).
A. A. Arsen’EV, “Ussr Comput. Math, and Math. Phys.”, 5, 110 (1965).
G. Scharf, “Helv. Phys. Acta”, 40, 929 (1967).
E. Hille and R. S. Phillips, “Functional Analysis and Semigroups”, American Mathematical Society Colloquium Publications, Providence, RI (1957).
T. Kato, “Perturbation Theory for Linear Operators”, Springer, Berlin (1966).
B. Fetz and S. F. Shen, in “Rarefied Gas Dynamics”, D. Dini, Ed., Vol. II, p. 729, Editrice Tecnico Scientifica, Pisa (1971).
F. A. GRüNbaum, in “The Boltzmann Equation”, F. A.Grünbaum, Ed., p. 103, Courant Institute, New York University, New York (1972).
J. T. Marti, “Nukleonik”, 8, 159 (1966).
J. Mika, “Nukleonik”, 9, 200 (1967).
J. Mika, “Nukleonik”, 9, 303 (1967).
S. Albertoni and B. Montagnini, in “Pulsed Neutron Research”, Vol. I, p. 239, Iaea, Vienna (1966).
S. Albertoni and B. Montagnini, “J. Math. Anal. Appl.”, 13, 19 (1966).
R. Bednarz, in “Pulsed Neutron Research”, Vol. I, p. 259, Iaea, Vienna (1966).
M. Ribaric, “Functional Analytic Concepts and Structures in Neutron Transport Theory”, Slovenska Akademya Znanosti in Unnetnosti, Ljubljana (1973).
H. G. Kaper, C. G. Lekkerkerker and J. Heitmanek, “Spectral Methods in Linear Transport Theory”, Birkhäuser, Basel (1982).
H. Grad, “Proc. Appl. Math.”, 17, 154 (1965).
S. Ukai, “Proc. Japan. Acad. Ser. A, Math. Sci.”, 50, 179 (1974).
T. Nishida and K. Imai, “Publ. Res. Inst Math. Sci, Kyoto Univ.”, 12, 229(1977).
Y. Shizuta and K. Asano, “Proc. Japan Acad, Ser. A, Math. Sci.”, 53, 3 (1977).
R. E. Calflisch, “Comm, Math. Phys.”, 74, 71 and 97 (1980).
Y. Shizuta, Preprint (1979).
R. Illner and M,. Shinbrot, “Comm. Math. Phys., 95, 217 (1984).
IC Hamdache, “Comptes Rendus”, Série I, 299, 431 (1984).
N. Bellomo and G. Toscani, “J. Math. Phys,”, 26, 334 (1985).
G. Toscani, “Arch. Rational Mech. Anal.”, 95, 37 (1986).
M. Shinbrot, “Transport Theory Statist. Phys,.”, 15, 317 (1986).
V. S. Galkin, “Prikl. Mat. Mekh.”, 20, 445 (1956).
V. S. Galkin, “Prikl. Mat. Mekh.”, 22, 532 (1958).
V. S. Galkin, “Prikl, Mat. Mekh.”, 28, 226 (1964).
V. S. Galkin, “Fluid Dynamics”, 1, 29 (1966).
C. Cercignani, submitted to “Arch. Rational Mech. Anal.” (1986).
D. R. Willis, in “Rarefied Gas Dynamics”, L. Talbot, Ed., p. 429, Academic Press, New York (1961).
H, Grad, “Siam J.”, 14, 935 (1966).
C Cercignani, “J. Math. Phys.”, 8, 1653 (1967).
C. Cercignani, “J. Math. Phys.”, 9, 633 (1968).
C. Cercignani, “Mathematical Methods in Kinetic Theory”, Plenum Press, New York (1969).
Y. P. Pao, “J. Math. Phys.”, 9, 1893 (1967).
J. P. Guiraud, “J. Méc. Théor. Appl.”, 7, 171 (1968).
J. P. Guiraud, “J. Méc. Théor. Appl.”, 9, 443 (1970).
J. P. Guiraud, “J. Méc. Théor. Appl.”, 11, 2 (1972).
J. P. Guiraud, “C.R. Acad. Sci. Paris”, 274, 417 (1972).
J. P. Guiraud, “C.R. Acad. Sci. Paris”, 275, 1259 (1972).
C. Rigolot-Turbat, “C.R. Acad. Sci. Paris”, 272, 617 (1971).
C. Rigolot-Turbat, “C.R. Acad. Sci. Paris”, 272, 763 (1971).
C. Rigolot-Turbat, “C.R. Acad. Sci. Paris”, 273, 58 (1971).
K. M. Case and P. F. Zweifel, “Linear Transport”, Addison-Wesley, Reading, MA. (1967).
S. Ukai and K. Asano, “Arch. Rational Mech. Anal.”, 84, 249 (1983).
C. Cercignani, in “Mathematical Problems on the Kinetic Theory of Gases”, D. C. Pack and H. Neunzert Eds., p. 129, Lang, Frankfurt (1980).
M. D. Arthur and C. Cercignani, “Z. Angew. Math. Phys.”, 31, 634 (1980).
C. E. Siewert and J. R. Thomas, “Z. Angew. Math. Phys.”, 32, 421 (1981).
C. E. Siewert and J. R. Thomas, “Z. Angew. Math. Phys”, 33, 202 (1982).
C. E. Siewert and J. R. Thomas, “Z. Angew. Math. Phys.”, 33, 626 (1982).
W. Greenberg and C. V. M. Van Der Mee, “Z. Angew. Math, Phys.”, 35, 156(1984).
C. Cercignani, in “Trends in Applications of Pure Mathematics to Mechanics”, E. Kröner and K. Kirchgässner, Eds., Lecture Notes in Physics, 249, p. 35 Springer, Berlin (1986).
C. Bardos, R. E. Caflisch and B. Nicolaenko, “Comm. Pure. Appl. Math.”, 39, 323(1986).
N. B. Maslova, “Ussr Comput. Math. Math. Phys.”, 22, 208 (1982).
C. Bardos, R. E. Caflisch and F. Golse (to appear, 1986).
C. Cercignani, R. Illner and M. Shinbrot “Duxe Math. J.” (Sept. 1987).
H. Schaefer, “Math. Ann.”, 129 (1955).
S. R. Smart, “Fixed Point Theorems”, Cambridge University Press, New York (1974).
S. Kawashima, in “Recent Topics in Nonlinear Pde (Hiroshima, 1983)”, p. 59, North-Holland Mathematical Studies, 98, North-Holland, Amsterdam (1984).
L. Arkeryd, R. Esposito and M. Pulvirenti (to appear in “Comm. Math. Phys.”, 1987).
R. Illner and M, Pulvirenti, “Comm. Math. Phys.”, 105, 189 (1986).
R. Illner and M. Pulvirenti (to appear in “Transport Theory Statist. Phys.”, 1987).
M. Pulvirenti (to appear in “Comm. Math. Phys.”, 1987).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media New York
About this chapter
Cite this chapter
Cercignani, C. (1988). Theorems on the Solutions of The Boltzmann Equation. In: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1039-9_8
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1039-9_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6995-3
Online ISBN: 978-1-4612-1039-9
eBook Packages: Springer Book Archive