# American Institute of Mathematical Sciences

November  2014, 19(9): 2739-2766. doi: 10.3934/dcdsb.2014.19.2739

## Transport semigroup associated to positive boundary conditions of unit norm: A Dyson-Phillips approach

 1 Università degli Studi di Udine, Dipartimento di Ingegneria Civile, via delle Scienze 208, 33100 Udine, Italy 2 Università degli Studi di Torino & Collegio Carlo Alberto, Department of Economics and Statistics, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy

Received  November 2013 Revised  May 2014 Published  September 2014

We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\mathcal{T}_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\mathcal{T}_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \geq 0}$ as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals $J \subseteq [0,\infty)$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.
Citation: Luisa Arlotti, Bertrand Lods. Transport semigroup associated to positive boundary conditions of unit norm: A Dyson-Phillips approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2739-2766. doi: 10.3934/dcdsb.2014.19.2739
##### References:
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##### References:
 [1] L. Arlotti, J. Banasiak and B. Lods, A new approach to transport equations associated to a regular field: trace results and well-posedness, Mediterr. J. Math., 6 (2009), 367-402. doi: 10.1007/s00009-009-0022-7.  Google Scholar [2] L. Arlotti, J. Banasiak and B. Lods, On general transport equations with abstract boundary conditions. The case of divergence free force field, Mediterr. J. Math., 8 (2011), 1-35. doi: 10.1007/s00009-010-0061-0.  Google Scholar [3] L. Arlotti, Explicit transport semigroup associated to abstract boundary conditions, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, I (2011), 102-111.  Google Scholar [4] L. Arlotti, Boundary conditions for streaming operator in a bounded convex body, Transp. Theory Stat. Phys., 15 (1986), 959-972. doi: 10.1080/00411458608212725.  Google Scholar [5] L. Arlotti and B. Lods, Substochastic semigroups for transport equations with conservative boundary conditions, J. Evolution Equations, 5 (2005), 485-508. doi: 10.1007/s00028-005-0209-8.  Google Scholar [6] L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend., 30 (2011), 457-495. doi: 10.4171/ZAA/1444.  Google Scholar [7] L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations, Commun. Pure Appl. Anal., 13 (2014), 729-771. doi: 10.3934/cpaa.2014.13.729.  Google Scholar [8] L. Arlotti, B. Lods and M. Mokhtar-Kharroubi, On perturbed substochastic once integrated semigroups in abstract state spaces,, in preparation., ().   Google Scholar [9] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, 2006.  Google Scholar [10] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation, application à l'équation de transport, Ann. Sci. École Norm. Sup., 3 (1970), 185-233.  Google Scholar [11] R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405. doi: 10.1016/0022-247X(87)90252-6.  Google Scholar [12] M. Boulanouar, New results in abstract time-dependent transport equations, Transport Theory Statist. Phys., 40 (2011), 85-125. doi: 10.1080/00411450.2011.603402.  Google Scholar [13] C. Cercignani, The Boltzmann Equation and its Applications, Springer Verlag, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar [14] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer Verlag, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar [15] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6: Evolution Problems II, Berlin, Springer, 2000. doi: 10.1007/978-3-642-58004-8.  Google Scholar [16] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New-York, 2000.  Google Scholar [17] B. Lods, Semigroup generation properties of streaming operators with noncontractive boundary conditions, Math. Comput. Modelling, 42 (2005), 1441-1462. doi: 10.1016/j.mcm.2004.12.007.  Google Scholar [18] M. Mokhtar-Kharroubi, On collisionless transport semigroups with boundary operators of norm one, J. Evolution Equations, 8 (2008), 327-362. doi: 10.1007/s00028-007-0360-5.  Google Scholar [19] M. Mokhtar-Kharroubi and J.Voigt, On honesty of perturbed substochastic $C_0$-semigroups in $L^1$-spaces, J. Operator Theory, 64 (2010), 131-147.  Google Scholar [20] J. Voigt, Functional Analytic Treatment of the Initial Boundary Value for Collisionless Gases, München, Habilitationsschrift, 1981. Google Scholar
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