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 & 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.  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.  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.  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.  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.  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).   Google Scholar

[7]

J. Banasiak and M. Lachowicz, Around the Kato generation theorem for semigroups,, Studia Math, 179 (2007), 217.  doi: 10.4064/sm179-3-2.  Google Scholar

[8]

J. Banasiak, Positivity in natural sciences,, in, (2008), 1.   Google Scholar

[9]

C. J. Batty and D. W. Robinson, Positive one-parameter semigroups on ordered Banach spaces,, Acta Appl. Math., 1 (1984), 221.  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, (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.  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, (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.  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, (2006).   Google Scholar

[16]

T. Kato, On the semi-groups generated by Kolmogoroff's differential equations,, J. Math. Soc. Jap., 6 (1954), 1.  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.  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.  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.   Google Scholar

[20]

M. Mokhtar-Kharroubi, New generation theorems in transport theory,, Afr. Mat., 22 (2011), 153.  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.  doi: 10.1007/s002330010006.  Google Scholar

[22]

B. de Pagter, Ordered Banach spaces,, in, (1987), 265.   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.  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.   Google Scholar

[25]

H. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation,, in, (2006), 135.   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.  doi: 10.1081/TT-100104455.  Google Scholar

[27]

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

[28]

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

[29]

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

[7]

J. Banasiak and M. Lachowicz, Around the Kato generation theorem for semigroups,, Studia Math, 179 (2007), 217.  doi: 10.4064/sm179-3-2.  Google Scholar

[8]

J. Banasiak, Positivity in natural sciences,, in, (2008), 1.   Google Scholar

[9]

C. J. Batty and D. W. Robinson, Positive one-parameter semigroups on ordered Banach spaces,, Acta Appl. Math., 1 (1984), 221.  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, (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.  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, (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.  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, (2006).   Google Scholar

[16]

T. Kato, On the semi-groups generated by Kolmogoroff's differential equations,, J. Math. Soc. Jap., 6 (1954), 1.  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.  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.  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.   Google Scholar

[20]

M. Mokhtar-Kharroubi, New generation theorems in transport theory,, Afr. Mat., 22 (2011), 153.  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.  doi: 10.1007/s002330010006.  Google Scholar

[22]

B. de Pagter, Ordered Banach spaces,, in, (1987), 265.   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.  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.   Google Scholar

[25]

H. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation,, in, (2006), 135.   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.  doi: 10.1081/TT-100104455.  Google Scholar

[27]

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

[28]

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

[29]

J. Voigt, On substochastic $C_0$-semigroups and their generators,, Transp. Theory. Stat. Phys, 16 (1987), 453.  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.  doi: 10.1007/BF02573236.  Google Scholar

[1]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[2]

Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907

[3]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[4]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[5]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[6]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[7]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[8]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[9]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

[10]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[11]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[12]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[13]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[14]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[15]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[16]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617

[17]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[18]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[19]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[20]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (1)

[Back to Top]