On the Asymptotic Spectrum of a Transport Operator with Elastic and Inelastic Collision Operators

Abstract

In this article, we investigate the spectral properties of a class of neutron transport operators involving elastic and inelastic collision operators introduced by Larsen and Zweifel [1]. Our analysis is manly focused on the description of the asymptotic spectrum which is very useful in the study of the properties of the solution to Cauchy problem governed by such operators (when it exists). The last section of this work is devoted to the properties of the leading eigenvalue (when it exists). So, we discuss the irreducibility of the semigroups generated by these operators. We close this section by discussing the strict monotonicity of the leading eigenvalue with respect to the parameters of the operator.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Larsen E W, Zweifel P F. On the spectrum of the linear transport operator. J Mathematical Phys, 1974, 15: 1987–1997

    MathSciNet  Article  Google Scholar 

  2. [2]

    Cercignani C, Ilner R, Pulvirenti M. The Mathematical Theory of Gases. New York: Springer Verlag, 1994

    Google Scholar 

  3. [3]

    Latrach K. On the spectrum of the transport operator with abstract boundary conditions in slab geometry. J Math Anal Appl, 2000, 252: 1–17

    MathSciNet  Article  Google Scholar 

  4. [4]

    Mokhtar-Kharroubi M. Mathematical topics in neutron transport theory. New aspects, Series on Advances in Mathematics for Applied Sciences 46. Singapor: World Scientific Publishing, 1997

    Google Scholar 

  5. [5]

    Mokhtar-Kharroubi M. Optimal spectral theory of the linear Boltzmann equations. J Funct Anal, 2005, 226: 21–47

    MathSciNet  Article  Google Scholar 

  6. [6]

    Sbihi M. Spectral theory of neutron transport semigroups with partly elastic collision operators. J Math Phys, 2006, 47: 123502 (12 pages)

    MathSciNet  Article  Google Scholar 

  7. [7]

    Sbihi M. Analyse Spectrale De Modèles Neutroniques. Besançon: Thèse de Doctorat de l’universit´e de Franche-Comté, 2005

    Google Scholar 

  8. [8]

    Ukai S. Eigenvalues of the neutron transport operator for a homogeneous finite moderator. J Math Ana Appl, 1967, 18: 297–314

    MathSciNet  Article  Google Scholar 

  9. [9]

    Vidav I. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J Math Anal Appl, 1968, 22: 144–155

    MathSciNet  Article  Google Scholar 

  10. [10]

    Vidav I. Spectra a perturbed semigroups with applications to transport theory. J Math Anal Appl, 1970, 30: 264–279

    MathSciNet  Article  Google Scholar 

  11. [11]

    Voigt J. Spectral properties of the neutron transport equation. J Math Anal App, 1985, 106: 14–153

    MathSciNet  Article  Google Scholar 

  12. [12]

    Weis L. A generalization of the Vidav-Jorgens perturbation theorem for semigroup and its application to transport theory. J Math Anal Appl, 1988, 129: 6–23

    MathSciNet  Article  Google Scholar 

  13. [13]

    Latrach K. Compactness results for transport equations and applications. Math Models Methods Appl Sci, 2001, 11: 1181–1202

    MathSciNet  Article  Google Scholar 

  14. [14]

    Beals R, Protopopescu V. Abstract time-dependent transport equations. J Math Anal Appl, 1987, 121: 370–405

    MathSciNet  Article  Google Scholar 

  15. [15]

    Cessenat M. Théorèmes de trace Lp pour des espaces de fonctions de la neutronique. C R Acad Sci Paris tome 299, 1984, 16(1): 831–834

    MATH  Google Scholar 

  16. [16]

    Cessenat M. Théorèmes de trace pour des espaces de fonctions de la neutronique. C R Acad Sci Paris tome 300, 1985, 3(1): 89–92

    MATH  Google Scholar 

  17. [17]

    Schechter M. Spectra of Partial Differential Operators. Amsterdam: North-Holland, 1971

    Google Scholar 

  18. [18]

    Marek I. Frobenius theory of positive operators: Comparison theorems and applications. SIAM J Appl Math, 1970, 19: 607–628

    MathSciNet  Article  Google Scholar 

  19. [19]

    Anselone P M, Palmer T W. Collectively compact sets of linear operators. Pacific J Math, 1968, 25: 417–422

    MathSciNet  Article  Google Scholar 

  20. [20]

    Yosida K. Functional Analysis. New York: Springer-Verlag, 1980

    Google Scholar 

  21. [21]

    Kosad Y, Latrach K. Regularity of the solution to the linear Boltzmann equation in finite bodies. J Math Anal Appl, 2017, 48(1): 506–537

    Article  Google Scholar 

  22. [22]

    Kaper H G, Lekkerkerker C G, Hejtmanek J. Spectral methods in linear transport theory//Operator Theory: Advances and Application Vol 5. Basel: Birkh¨auser, 1982

  23. [23]

    Reed M, Simon B. Methods of modern mathematical physics. I. Functional analysis. New York-London: Academic Press, 1972

    Google Scholar 

  24. [24]

    Mayer-Niberg P. Banach Lattices. New York: Springer Verlag, 2001

    Google Scholar 

  25. [25]

    Lods B. On linear Kinetic equations involving unbounded cross-section. Math Methods Appl Sci, 2004, 27: 1049–1075

    MathSciNet  Article  Google Scholar 

  26. [26]

    Takac P. A spectral mapping theorem for the exponential function in linear transport theory. Transp Theory Stat Phys, 1985, 14: 655–667

    MathSciNet  Article  Google Scholar 

  27. [27]

    Dunford N, Schwartz J T. Linear Operators: Part I. New York: Intersciences, 1958

    Google Scholar 

  28. [28]

    Nagel R. One-parameter Semigroups of Positive Operators//Lecture Notes Math, 1184. New York: Springer Verlag, 1986

    Google Scholar 

  29. [29]

    Kato T. Perturbation Theory for Linear Operators. New York: Springer-Verlag, 1966

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Khalid Latrach.

Electronic supplementary material

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Al-Izeri, AM., Latrach, K. On the Asymptotic Spectrum of a Transport Operator with Elastic and Inelastic Collision Operators. Acta Math Sci 40, 805–823 (2020). https://doi.org/10.1007/s10473-020-0315-2

Download citation

Keywords

  • Compactness properties
  • transport operator
  • abstract boundary conditions
  • asymptotic spectrum
  • irreducibility
  • leading eigenvalue

2010 MR Subject Classification

  • 47A10
  • 47A55
  • 35Q20