×

A compactness result for perturbed semigroups and application to a transport model. (English) Zbl 1180.47028

The authors investigate the remainder term of order \(n\), \(R_n(t)=\sum_{j=n}^{\infty}U_j(t)\), of the Dyson-Phillips expansion \(V(t)=\sum_{j=0}^{\infty}U_j(t)\) of the strongly continuous semigroup \((V(t))_{t>0}\) generated by the operator \(T+K\), where \(T\) is the generator of a strongly continuous semigroup \((U(t))_{t\geq 0}\) on a Banach space \(X\) and \(K\) is a bounded linear operator in \(X\). Here, \((U_j(t))_{j\geq 0}\) is defined by \(U_0(t)=U(t)\) and \(U_j(t)=\int_0^tU(s)KU_{j-1}(t-s)\,ds\) for \(j\geq 1\).
The main result of the paper shows that, under some assumptions, \(R_{2m+1}(t)\) is compact on \(X\) for each \(t>0\), and therefore \(U(t)\) and \(V(t)\) have the same essential type. This theorem is then applied to the study of the time asymptotic behaviour of the solution of a one-dimensional transport equation with reentry boundary conditions on \(L_1\)-spaces without regularity conditions on the initial data.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47N50 Applications of operator theory in the physical sciences
82C70 Transport processes in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brendle, S., On the asymptotic behavior of perturbed strongly continuous semigroups, Math. Nachr., 226, 35-47 (2001) · Zbl 0986.47034
[2] Greenberg, W.; Van der Mee, C.; Protopopescu, V., Boundary Value Problems in Abstract Kinetic Theory (1987), Birkhäuser · Zbl 0624.35003
[3] Hille, H.; Phillips, R. E., Functional Analysis and Semigroups, Amer. Math. Soc. Colloq., vol. 31 (1957), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[4] Latrach, K.; Lods, B., Regularity and time asymptotic behaviour of solutions to transport equations, Transport Theory Statist. Phys., 30, 617-639 (2001) · Zbl 0990.82029
[5] K. Latrach, B. Lods, Spectral analysis of transport equations with bounce-back boundary conditions, Math. Methods Appl. Sci., in press; K. Latrach, B. Lods, Spectral analysis of transport equations with bounce-back boundary conditions, Math. Methods Appl. Sci., in press · Zbl 1163.47034
[6] Lods, B.; Sbihi, M., Stability of the essential spectrum for \(2-D\)-transport models with maxwell boundary conditions, Math. Methods Appl. Sci., 29, 499-523 (2006) · Zbl 1092.35057
[7] Mokhtar-Kharroubi, M., Time asymptotic behaviour and compactness in neutron transport theory, Eur. J. Mech. B Fluids, 11, 39-68 (1992)
[8] Mokhtar-Kharroubi, M., Mathematical Topics in Neutron Transport Theory, New Aspects, Ser. Adv. Math. Appl. Sci., vol. 46 (1997), World Scientific · Zbl 0997.82047
[9] Mokhtar-Kharroubi, M., Optimal spectral theory of the linear Boltzmann equation, J. Funct. Anal., 226, 21-47 (2005) · Zbl 1088.47033
[10] Sbihi, M., A resolvent approach to the stability of essential and critical spectra of perturbed \(C_0\)-semigroups on Hilbert spaces with applications to transport theory, J. Evol. Equ., 7, 35-58 (2007) · Zbl 1117.35056
[11] Song, D., Some notes on the spectral properties of \(C_0\)-semigroups generated by linear transport operators, Transport Theory Statist. Phys., 26, 233-242 (1997) · Zbl 0915.47030
[12] Song, D.; Greenberg, W., Spectral properties of transport equations for slab geometry in \(L^1\) with reentry boundary conditions, Transport Theory Statist. Phys., 30, 325-355 (2001) · Zbl 1009.47036
[13] Song, D.; Wang, M.; Zhu, G., Asymptotic expansion and asymptotic behavior of the solution for the time dependent neutron transport problem in a slab with generalized boundary conditions, Sys. Sci. Math. Sci., 3, 102-125 (1990) · Zbl 0734.45008
[14] Takac, P., A spectral mapping theorem for the exponential function in linear transport theory, Transport Theory Statist. Phys., 14, 655-667 (1985) · Zbl 0613.45014
[15] Vidav, I., Existence and uniqueness of nonnegative eigenfunction of the Boltzmann operator, J. Math. Anal. Appl., 22, 144-155 (1968) · Zbl 0155.19203
[16] Vidav, I., Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30, 264-279 (1970) · Zbl 0195.13704
[17] Voigt, J., A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90, 153-161 (1980) · Zbl 0433.47022
[18] Voigt, J., Spectral properties of the neutron transport equation, J. Math. Anal. Appl., 106, 140-153 (1985) · Zbl 0567.45002
[19] Voigt, J., On the convex compactness property of the strong operator topology, Note Mat., 12, 259-269 (1992) · Zbl 0804.46012
[20] Weis, L., A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory, J. Math. Anal. Appl., 129, 6-23 (1988) · Zbl 0648.47015
[21] Weis, L., The stability of positive semigroups on \(L_p\) spaces, Proc. Amer. Math. Soc., 123, 3089-3094 (1995) · Zbl 0851.47028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.