# American Institute of Mathematical Sciences

March  2014, 13(2): 729-771. doi: 10.3934/cpaa.2014.13.729

## 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

Received  March 2013 Revised  August 2013 Published  October 2013

We provide a honesty theory of substochastic evolution families in real abstract state space, extending to an non-autonomous setting the result obtained for $C_0$-semigroups in our recent contribution [On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend. 30, 457--495, 2011]. The link with the honesty theory of perturbed substochastic semigroups is established. Application to non-autonomous linear Boltzmann equation is provided.
Citation: Luisa Arlotti, Bertrand Lods, Mustapha Mokhtar-Kharroubi. Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations. Communications on Pure and Applied Analysis, 2014, 13 (2) : 729-771. doi: 10.3934/cpaa.2014.13.729
##### 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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##### 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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