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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

Luisa Arlotti 1, , Bertrand Lods 2, and  Mustapha Mokhtar-Kharroubi 3,

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.
Keywords: Substochastic semigroups, evolution family, additive norm, linear Boltzmann equation..
Mathematics Subject Classification: Primary: 47D06; Secondary: 47D30, 47D07, 47N5.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

J. Banasiak, Positivity in natural sciences, in "Multiscale Problems in the Life Sciences," Lecture Notes in Math., 1940, Springer, Berlin, (2008), 1-89. Google Scholar

[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.  Google Scholar

[10]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical surveys and monographs 70, AMS, 1999. Google Scholar

[11]

E. B. Davies, "Quantum Theory of Open Systems," Academic Press, 1976. Google Scholar

[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.  Google Scholar

[13]

K. J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer, New-York, 2000. Google Scholar

[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.  Google Scholar

[15]

A. Gulisashvili and J. A. van Casteren, "Non-autonomous Kato Classes and Feynman-Kac Propagators," World Scientific, Singapore, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

M. Mokhtar-Kharroubi, New generation theorems in transport theory, Afr. Mat., 22 (2011), 153-176. doi: 10.1007/s13370-011-0014-1.  Google Scholar

[21]

S. Monniaux and A. Rhandi, Semigroup methods to solve non-autonomous evolution equations, Semigroup Forum, 60 (2000), 122-134. doi: 10.1007/s002330010006.  Google Scholar

[22]

B. de Pagter, Ordered Banach spaces, in "One-parameter Semigroups" (Ph. Clément ed.), North-Holland, Amserdam, (1987), 265-279. Google Scholar

[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.  Google Scholar

[24]

F. Räbiger, R. Schnaubelt, A. Rhandi and J. Voigt, Non-autonomous Miyadera perturbations, Differential Integral Equations, 13 (2000), 341-368.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229 (1977), 163-171. doi: 10.1007/BF01351602.  Google Scholar

[28]

J. Voigt, "Functional Analytic Treatment of the Initial Boundary Value Problem for Collisionless Gases," Habilitationsschrift, München, 1981. Google Scholar

[29]

J. Voigt, On substochastic $C_0$-semigroups and their generators, Transp. Theory. Stat. Phys, 16 (1987), 453-466. doi: 10.1080/00411458708204302.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

J. Banasiak, Positivity in natural sciences, in "Multiscale Problems in the Life Sciences," Lecture Notes in Math., 1940, Springer, Berlin, (2008), 1-89. Google Scholar

[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.  Google Scholar

[10]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical surveys and monographs 70, AMS, 1999. Google Scholar

[11]

E. B. Davies, "Quantum Theory of Open Systems," Academic Press, 1976. Google Scholar

[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.  Google Scholar

[13]

K. J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer, New-York, 2000. Google Scholar

[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.  Google Scholar

[15]

A. Gulisashvili and J. A. van Casteren, "Non-autonomous Kato Classes and Feynman-Kac Propagators," World Scientific, Singapore, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

M. Mokhtar-Kharroubi, New generation theorems in transport theory, Afr. Mat., 22 (2011), 153-176. doi: 10.1007/s13370-011-0014-1.  Google Scholar

[21]

S. Monniaux and A. Rhandi, Semigroup methods to solve non-autonomous evolution equations, Semigroup Forum, 60 (2000), 122-134. doi: 10.1007/s002330010006.  Google Scholar

[22]

B. de Pagter, Ordered Banach spaces, in "One-parameter Semigroups" (Ph. Clément ed.), North-Holland, Amserdam, (1987), 265-279. Google Scholar

[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.  Google Scholar

[24]

F. Räbiger, R. Schnaubelt, A. Rhandi and J. Voigt, Non-autonomous Miyadera perturbations, Differential Integral Equations, 13 (2000), 341-368.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229 (1977), 163-171. doi: 10.1007/BF01351602.  Google Scholar

[28]

J. Voigt, "Functional Analytic Treatment of the Initial Boundary Value Problem for Collisionless Gases," Habilitationsschrift, München, 1981. Google Scholar

[29]

J. Voigt, On substochastic $C_0$-semigroups and their generators, Transp. Theory. Stat. Phys, 16 (1987), 453-466. doi: 10.1080/00411458708204302.  Google Scholar

[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.  Google Scholar

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