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compactness; essential type; positivity and irreducibility; spectral properties; streaming operator; strongly continuous semigroups
This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator $K$. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un $C_0$-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case $\Vert K\Vert \ge 1$. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.
[1] F. Ammar-Khodja, M.  Mokhtar-Kharroubi: On the exponential stability of advection semigroups with boundary operator. Math. Models Methods Appl. Sci. 8 (1998), 95–106. DOI 10.1142/S0218202598000056 | MR 1612007
[2] R. Beals, V.  Protopopescu: Abstract time-dependent transport equations. J.  Math. Anal. Appl. 121 (1987), 370–405. DOI 10.1016/0022-247X(87)90252-6 | MR 0872231
[3] G. Borgioli, S.  Totaro: Semigroup generation properties of the streaming operator in dependence of the boundary conditions. Transp. Theory Stat. Phys. 25 (1996), 491–502. DOI 10.1080/00411459608220716 | MR 1407549
[4] G. Borgioli, S. Totaro: 3D-streaming operator with multiplying boundary conditions: Semigroup generation properties. Semigroup Forum 55 (1997), 110–117. DOI 10.1007/PL00005905 | MR 1446663
[5] M. Boulanouar: L’opérateur d’Advection: existence d’un $C_0$-semi-groupe (I). Transp. Theory Stat. Phys. 31 (2002), 153–167. MR 1904837
[6] Ph. Clément, H. J. A. M. Hijmans, C. J. van Duijn, B. de Paqter: One-Parameter Semigroups. North-Holland, Amsterdam-New York, 1987. MR 0915552
[7] W. Greenberg, C. van der Mee, V.  Protopopescu: Boundary Value Problems in Abstract Kinetic Theory. Birkhäuser, Basel, 1987. MR 0896904
[8] R. Dautray, J.-L.  Lions: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Vol. 9: Évolution: numérique, transport. Masson, Paris, 1988. MR 1016606
[9] One-Parameter Semigroups of Positive operators. Lecture Notes in Mathematics 1184. R. Nagel (ed.), Springer, Berlin-New York, 1986. MR 0839450
[10] B. de Paqter: Irreducible compact operators. Math.  Z. 192 (1986), 149–153. DOI 10.1007/BF01162028 | MR 0835399
[11] M. Schechter: Spectra of Partial Differential Operators. North-Holland, Amsterdam, 1971. MR 0869254 | Zbl 0225.35001
[12] R. Sentis: Equation de transport avec des conditions aux limites de type réflexion. Rapport de recherche, INRIA no. 162, Le Chesnay (France).
[13] S. Ukai: Solutions of the Boltzmann equation. Patterns and waves. Stud. Math. Appl. 18 (1996), 37–96. MR 0882376
[14] J. Voigt: Functional analytic treatment of the initial boundary value problem for collisionless gases. Habilitationsschrift, Universität München, 1981.
[15] L. Weis: The stability of positive semigroups on $L_p$-spaces. Proc. Am. Mat. Soc. 123 (1995), 3089–3094. MR 1273529
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