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Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations
1. | Dipartimento di Ingegneria Civile, Università di Udine, via delle Scienze 208, 33100 Udine, Italy |
2. | Università degli, Studi di Torino and Collegio Carlo Alberto, Department of Statistics and Economics, Corso Unione Sovietica, 218/bis,, 10134 Torino, Italy |
3. | Université de Franche–Comté, Laboratoire de Mathématiques, CNRS UMR 6623, 16, route de Gray, 25030 Besançon Cedex |
References:
[1] |
L. Arlotti, The Cauchy problem for the linear Maxwell-Bolztmann equation, J. Differential Equations, 69 (1987), 166-184.
doi: 10.1016/0022-0396(87)90115-X. |
[2] |
L. Arlotti, A perturbation theorem for positive contraction semigroups on $L^1$-spaces with applications to transport equation and Kolmogorov's differential equations, Acta Appl. Math., 23 (1991), 129-144.
doi: 10.1007/BF00048802. |
[3] |
L. Arlotti and J. Banasiak, Strictly substochastic semigroups with application to conservative and shattering solution to fragmentation equation with mass loss, J. Math. Anal. Appl., 293 (2004), 673-720.
doi: 10.1016/j.jmaa.2004.01.028. |
[4] |
L. Arlotti and J. Banasiak, Nonautonomous fragmentation equation via evolution semigroups, Math. Meth. Appl. Sci., 33 (2010), 1201-1210.
doi: 10.1002/mma.1282. |
[5] |
L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend., 30 (2011), 457-495.
doi: 0.4171/ZAA/1444. |
[6] |
L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations, preprint, 2013, http://arxiv.org/abs/1303.7100. |
[7] |
J. Banasiak and M. Lachowicz, Around the Kato generation theorem for semigroups, Studia Math, 179 (2007), 217-238.
doi: 10.4064/sm179-3-2. |
[8] |
J. Banasiak, Positivity in natural sciences, in "Multiscale Problems in the Life Sciences," Lecture Notes in Math., 1940, Springer, Berlin, (2008), 1-89. |
[9] |
C. J. Batty and D. W. Robinson, Positive one-parameter semigroups on ordered Banach spaces, Acta Appl. Math., 1 (1984), 221-296.
doi: 10.1007/BF02280855. |
[10] |
C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical surveys and monographs 70, AMS, 1999. |
[11] |
E. B. Davies, "Quantum Theory of Open Systems," Academic Press, 1976. |
[12] |
E. B. Davies, Quantum dynamical semigroups and the neutron diffusion equation, Rep. Math. Phys., 11 (1977), 169-188.
doi: 10.1016/0034-4877(77)90059-3. |
[13] |
K. J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer, New-York, 2000. |
[14] |
G. Frosali, C. van der Mee and F. Mugelli, A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators, Math. Meth. Appl. Sci., 27 (2004), 669-685.
doi: 10.1002/mma.495. |
[15] |
A. Gulisashvili and J. A. van Casteren, "Non-autonomous Kato Classes and Feynman-Kac Propagators," World Scientific, Singapore, 2006. |
[16] |
T. Kato, On the semi-groups generated by Kolmogoroff's differential equations, J. Math. Soc. Jap., 6 (1954), 1-15.
doi: 10.2969/jmsj/00610001. |
[17] |
V. Liskevich, H. Vogt and J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal., 238 (2006), 245-277.
doi: 10.1016/j.jfa.2006.04.010. |
[18] |
M. Mokhtar-Kharroubi, On perturbed positive $C_0$-semigroups on the Banach space of trace class operators, Infinite Dim. Anal. Quant. Prob. Related Topics, 11 (2008), 1-21.
doi: 10.1142/S0219025708003130. |
[19] |
M. Mokhtar-Kharroubi and J. Voigt, On honesty of perturbed substochastic $C_0$-semigroups in $L^1$-spaces, J. Operator Th, 64 (2010), 101-117. |
[20] |
M. Mokhtar-Kharroubi, New generation theorems in transport theory, Afr. Mat., 22 (2011), 153-176.
doi: 10.1007/s13370-011-0014-1. |
[21] |
S. Monniaux and A. Rhandi, Semigroup methods to solve non-autonomous evolution equations, Semigroup Forum, 60 (2000), 122-134.
doi: 10.1007/s002330010006. |
[22] |
B. de Pagter, Ordered Banach spaces, in "One-parameter Semigroups" (Ph. Clément ed.), North-Holland, Amserdam, (1987), 265-279. |
[23] |
F. Räbiger, A. Rhandi and R. Schnaubelt, Perturbation and an abstract characterization of evolution semigroups, J. Math. Anal. Appl., 198 (1996), 516-533.
doi: 10.1006/jmaa.1996.0096. |
[24] |
F. Räbiger, R. Schnaubelt, A. Rhandi and J. Voigt, Non-autonomous Miyadera perturbations, Differential Integral Equations, 13 (2000), 341-368. |
[25] |
H. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, in "Proceedings Positivity IV- Theory and Applications," Dresden (Germany), (2006), 135-146. |
[26] |
C. van der Mee, Time-dependent kinetic equations with collision terms relatively bounded with respect to the collision frequency, Transport Theory and Statistical Physics, 30 (2001), 63-90.
doi: 10.1081/TT-100104455. |
[27] |
J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229 (1977), 163-171.
doi: 10.1007/BF01351602. |
[28] |
J. Voigt, "Functional Analytic Treatment of the Initial Boundary Value Problem for Collisionless Gases," Habilitationsschrift, München, 1981. |
[29] |
J. Voigt, On substochastic $C_0$-semigroups and their generators, Transp. Theory. Stat. Phys, 16 (1987), 453-466.
doi: 10.1080/00411458708204302. |
[30] |
J. Voigt, On resolvent positive operators and positive $C_0$-semigroups on $AL$-spaces, Semigroup Forum, 38 (1989), 263-266.
doi: 10.1007/BF02573236. |
show all references
References:
[1] |
L. Arlotti, The Cauchy problem for the linear Maxwell-Bolztmann equation, J. Differential Equations, 69 (1987), 166-184.
doi: 10.1016/0022-0396(87)90115-X. |
[2] |
L. Arlotti, A perturbation theorem for positive contraction semigroups on $L^1$-spaces with applications to transport equation and Kolmogorov's differential equations, Acta Appl. Math., 23 (1991), 129-144.
doi: 10.1007/BF00048802. |
[3] |
L. Arlotti and J. Banasiak, Strictly substochastic semigroups with application to conservative and shattering solution to fragmentation equation with mass loss, J. Math. Anal. Appl., 293 (2004), 673-720.
doi: 10.1016/j.jmaa.2004.01.028. |
[4] |
L. Arlotti and J. Banasiak, Nonautonomous fragmentation equation via evolution semigroups, Math. Meth. Appl. Sci., 33 (2010), 1201-1210.
doi: 10.1002/mma.1282. |
[5] |
L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend., 30 (2011), 457-495.
doi: 0.4171/ZAA/1444. |
[6] |
L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations, preprint, 2013, http://arxiv.org/abs/1303.7100. |
[7] |
J. Banasiak and M. Lachowicz, Around the Kato generation theorem for semigroups, Studia Math, 179 (2007), 217-238.
doi: 10.4064/sm179-3-2. |
[8] |
J. Banasiak, Positivity in natural sciences, in "Multiscale Problems in the Life Sciences," Lecture Notes in Math., 1940, Springer, Berlin, (2008), 1-89. |
[9] |
C. J. Batty and D. W. Robinson, Positive one-parameter semigroups on ordered Banach spaces, Acta Appl. Math., 1 (1984), 221-296.
doi: 10.1007/BF02280855. |
[10] |
C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical surveys and monographs 70, AMS, 1999. |
[11] |
E. B. Davies, "Quantum Theory of Open Systems," Academic Press, 1976. |
[12] |
E. B. Davies, Quantum dynamical semigroups and the neutron diffusion equation, Rep. Math. Phys., 11 (1977), 169-188.
doi: 10.1016/0034-4877(77)90059-3. |
[13] |
K. J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer, New-York, 2000. |
[14] |
G. Frosali, C. van der Mee and F. Mugelli, A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators, Math. Meth. Appl. Sci., 27 (2004), 669-685.
doi: 10.1002/mma.495. |
[15] |
A. Gulisashvili and J. A. van Casteren, "Non-autonomous Kato Classes and Feynman-Kac Propagators," World Scientific, Singapore, 2006. |
[16] |
T. Kato, On the semi-groups generated by Kolmogoroff's differential equations, J. Math. Soc. Jap., 6 (1954), 1-15.
doi: 10.2969/jmsj/00610001. |
[17] |
V. Liskevich, H. Vogt and J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal., 238 (2006), 245-277.
doi: 10.1016/j.jfa.2006.04.010. |
[18] |
M. Mokhtar-Kharroubi, On perturbed positive $C_0$-semigroups on the Banach space of trace class operators, Infinite Dim. Anal. Quant. Prob. Related Topics, 11 (2008), 1-21.
doi: 10.1142/S0219025708003130. |
[19] |
M. Mokhtar-Kharroubi and J. Voigt, On honesty of perturbed substochastic $C_0$-semigroups in $L^1$-spaces, J. Operator Th, 64 (2010), 101-117. |
[20] |
M. Mokhtar-Kharroubi, New generation theorems in transport theory, Afr. Mat., 22 (2011), 153-176.
doi: 10.1007/s13370-011-0014-1. |
[21] |
S. Monniaux and A. Rhandi, Semigroup methods to solve non-autonomous evolution equations, Semigroup Forum, 60 (2000), 122-134.
doi: 10.1007/s002330010006. |
[22] |
B. de Pagter, Ordered Banach spaces, in "One-parameter Semigroups" (Ph. Clément ed.), North-Holland, Amserdam, (1987), 265-279. |
[23] |
F. Räbiger, A. Rhandi and R. Schnaubelt, Perturbation and an abstract characterization of evolution semigroups, J. Math. Anal. Appl., 198 (1996), 516-533.
doi: 10.1006/jmaa.1996.0096. |
[24] |
F. Räbiger, R. Schnaubelt, A. Rhandi and J. Voigt, Non-autonomous Miyadera perturbations, Differential Integral Equations, 13 (2000), 341-368. |
[25] |
H. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, in "Proceedings Positivity IV- Theory and Applications," Dresden (Germany), (2006), 135-146. |
[26] |
C. van der Mee, Time-dependent kinetic equations with collision terms relatively bounded with respect to the collision frequency, Transport Theory and Statistical Physics, 30 (2001), 63-90.
doi: 10.1081/TT-100104455. |
[27] |
J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229 (1977), 163-171.
doi: 10.1007/BF01351602. |
[28] |
J. Voigt, "Functional Analytic Treatment of the Initial Boundary Value Problem for Collisionless Gases," Habilitationsschrift, München, 1981. |
[29] |
J. Voigt, On substochastic $C_0$-semigroups and their generators, Transp. Theory. Stat. Phys, 16 (1987), 453-466.
doi: 10.1080/00411458708204302. |
[30] |
J. Voigt, On resolvent positive operators and positive $C_0$-semigroups on $AL$-spaces, Semigroup Forum, 38 (1989), 263-266.
doi: 10.1007/BF02573236. |
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