Abstract
The time dependent transport equation in a sphere with reflecting boundary conditions is discussed in the setting of L 2. Some aspects of the spectral properties of the strongly continuous semigroup T(t) generated by the corresponding transport operator A are studied, and it is shown that the spectrum of T(t) outside the disk {λ: |λ| ≤ exp(−λ*t)} (where λ* is the essential infimum of the total collision frequency σ (r, v), or λ* = ess inf r lim v →0+ σ (r, v)) consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of σ(T(t))∩{λ : |λ| > exp(−λ*t)} can only appear on the circle {λ : |λ| = exp(−λ*t)}. Consequently, the asymptotic behavior of the time dependent solution is obtained.
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References
G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Math. Z. 185, 167–177 (1984).
G. Greiner and R. Nagel, On stability of strongly continuous semigroups of positive operators on L2(µ), Ann. Scuola Norm. Sup. Pisa 10, 257–262 (1983).
Falun Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs. 11, 43–56 (1985).
K. Jörgens, An asymptotic expansion in the theory of neutron transport, Comm. Pure Appl. Math. 11, 219–242 (1958).
J. Lehner and G. M. Wing, Solution of the linearized Boltzmann transport equation for slab geometry, Duke Math. J. 23, 125–142 (1956).
Peng Lei and Mingzhu Yang, On the spectrum of the transport operator in a homogeneous sphere with partly reflection boundary conditions, Kexue Tongbao 31: 24, 1867–1871 (1986) (in Chinese).
E. W. Larsen and P. F. Zweifel, On the spectrum of the linear transport operator, J. Math. Phys. 15, 1987–1997 (1974).
M. Mokhtar-Kharroubi, Time asymptotic behavior and compactness in transport theory, Eur. J. Mech. B: Fluids 11, 39–68 (1992).
R. Nagel ed., One-Parameter Semigroups of Positive Operators, (Lect. Notes in Math. 1184, Springer-Verlag, Berlin, 1986 ).
J. Pruss, On the spectrum of Co semigroups, Transl. Amer. Math. Soc. 284: 2, 847–857 (1984).
D. C. Salmi, N. S. Gads and N. G. Sjostrand, Spectrum of one-speed neutron transport operator with reflective boundary conditions in slab geometry, Transport Theory and Statistical Physics 24, 629–656 (1995).
Degong Song, Miansen Wang and Guangtian Zhu, A note on the spectrum of the transport operator with continuous energy, Transp. Theory Stat. Phys. 22: 5, 709–721 (1993).
R. van Norton, On the real spectrum of a mono-energetic neutron transport operator, Comm. Pure Appl. Math. 15, 149–158 (1962).
I. Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl. 30, 264–279 (1970).
J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Mh. Math. 90, 153–161 (1980).
J. Voigt, Spectral properties of the neutron transport equation, J. Math. Anal. Appl. 106, 140–153 (1985).
Haiyan Wang, Mingzhu Yang and Wenwei Wang, On the spectrum of a spherical symmetric transport operator with generalized boundary conditions, J. Southeast Univ. 19: 5, 7–17 (1989) (in Chinese).
L. W. Weis, A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory, J. Math. Anal. Appl. 129, 6–23 (1988).
Xianwen Zhang and Benzhong Liang, On the spectrum of a one-velocity transport operator with Maxwell boundary condition, J. Math. Anal. Appl. 202: 3, 920–939 (1996).
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Song, D., Greenberg, W. (2001). Asymptotics of Transport Equations for Spherical Geometry in L2 with Reflecting Boundary Conditions. In: Uvarova, L.A., Latyshev, A.V. (eds) Mathematical Modeling. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3397-6_19
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DOI: https://doi.org/10.1007/978-1-4757-3397-6_19
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