Abstract
The linearized relativistic Boltzmann equation inL 2(r,p) is investigated. The detailed analysis of the collision operatorL is carried out for a wide class of scattering cross sections.L is proved to have a form of the multiplication operatorv(p) plus the compact inL 2(p) perturbationK. The collisional frequencyv(p) is analysed to discriminate between relativistic soft and hard interactions. Finally, the existence and uniqueness of the solution to the linearized relativistic Boltzmann equation is proved.
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de Groot, S.R., van Leeuwen, W.A., van Weert, Ch.G.: Relativistic kinetic theory-principles and applications. Amsterdam: North-Holland 1980
Stewart, J.M.: Non-equilibrium relativistic kinetic theory. In: Lecture Notes in Physics, Vol. 10. Berlin, Heidelberg, New York: Springer 1971
Bichteler, K.: On the Cauchy problem of the relativistic Boltzmann equation. Commun. Math. Phys.4, 352–364 (1967)
Bancel, D.: Problème de Cauchy pour l'équation de Boltzmann en relativité générale. Ann. Inst. Henri Poincaré18, 263–284 (1973)
Bancel, D., Choquet-Bruhat, Y.: Existence, uniqueness and local stability for the Einstein-Maxwell-Boltzmann system. Commun. Math. Phys.33, 83–96 (1973)
Grad, H.: In: Third International Rarefield Gas Symposium. Vol. 1, p. 26. Lauremann, J.A. (ed.). New York: Academic Press 1963
Grad, H.: Asymptotic theory of the Boltzmann equation. Phys. Fluids6, 147–181 (1963);
Grad, H.: Asymptotic equivalence of the Navier-Stokes and non-linear Boltzmann equations. Proc. Symp. Appl. Math., Am. Math. Soc.17, 154–183 (1965)
Ellis, R.S., Pinsky, M.A.: The first and the second fluid approximations to the linearized Boltzmann equation. J. Math. Pures. Appl.54, 125–156 (1975);
Ellis, R.S., Pinsky, M.A.: The projection of the Navier-Stokes equations upon the Euler equations. J. Math. Pures. Appl.54, 157–182 (1975)
Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys.61, 119–148 (1978)
Kawashima, S., Matsumara, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equations. Commun. Math. Phys.70, 97–124 (1979)
Jutner, F.: Ann. Phys. (Leipzig)34, 856 (1911);35, 145 (1911)
Abramovitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs and mathematical tables. New York: Dover 1965
Ekiel-Jeżewska, M.L.: In: Recent developments in non-equilibrium thermodynamics. In: Lecture Notes in Physics, Vol. 199. Berlin, Heidelberg, New York: Springer 1984
van Weert, Ch.G.: Generalized hydrodynamics from relativistic kinetic theory. Physica.111A, 537–552 (1982)
Drange, H.B.: The linearized Boltzmann collision operator for cut-off potentials. SIAM J. Appl. Math.29, 665–676 (1975)
Cercignani, C.: Mathematical methods in kinetic theory. New York: Plenum/London, McMillan 1969
Cercignani, C.: Theory and application of the Boltzmann equation. Edinburgh: Scottish Academic Press; New York: Elsevier 1975
Grad, H.: Rarefied gas dynamics. London, Oxford, New York: Pergamon Press 1960
Dijkstra, J.J., van Leeuwen, W.A.: Mathematical aspects of relativistic kinetic theory. Physica90A, 450–486 (1978)
Hutson, V.C.L., Pym, J.S.: Applications of functional analysis and operator theory. London, New York: Academic Press 1980
Prudnikov, A.P., Britchkov, Yu.A., Maritchev, O.I.: Intiegraly i Riady, Specjalnyje Funkcji. Moskva: Nauka 1983 (in Russian)
Caflisch, R.E.: The Boltzmann equation with a soft potential. Commun. Math. Phys.74, 71–95 (1980) and74, 97–109 (1980)
Schechter, M.: On the essential spectrum of an arbitrary operator. J. Math. Anal. Appl.13, 205–215 (1966)
Ukai, S.: On the existence of global solutions of mixed problem for nonlinear Boltzmann equation. Proc. Jpn. Acad.50, 179–184 (1974)
Nishida, T., Imai, L.: Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. RIMS Kyoto,12, 229–239 (1976)
Adams, R.A.: Sobolev spaces. New York: Academic Press 1975
Dudyński, M., Ekiel-Jeżewska, M.L.: Causality of the linearized relativistic Boltzmann equation. Phys. Rev. Lett.55, 2831–2834 (1985)
Dudyński, M., Ekiel-Jeżewska, M.L.: Causality problem in the relativistic kinetic theory. In: Recent developments in nonequilibrium thermodynamics. Lecture Notes in Physics, Vol. 253. Berlin, Heidelberg, New York: Springer 1986
Dudyński, M., Ekiel-Jeżewska, M.L.: Relativistic Boltzmann equation — mathematical and physical aspects. In: Relativistic fluid dynamics. Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer (to appear)
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Communicated by J. L. Lebowitz
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Dudyński, M., Ekiel-Jeżewska, M.L. On the linearized relativistic Boltzmann equation. Commun.Math. Phys. 115, 607–629 (1988). https://doi.org/10.1007/BF01224130
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DOI: https://doi.org/10.1007/BF01224130