\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 47D06; Secondary: 47D30, 47D07, 47N50.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(60) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return