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Spectral properties and time asymptotic behaviour of linear transport equations in slab geometry. (English) Zbl 1103.82027

The authors consider a one-dimensional transport equation in a slab, whose abstract form reads as follows: \[ \dot \psi(t) = \big( T_H + K) \psi(t), \qquad \psi^-(t) = H\psi^+(t). \] Here \(K\) is a bounded perturbation, accounting for scattering phenomena, \(T_H\) is a unbounded operator, accounting for streaming and absorption, and \(H\) is a boundary operator relating the inflow \(\psi^-\) with the outflow \(\psi^+\). The functional setting of the evolution problem is that of \(L^p\) functions on the phase-space, with \(p \in [1,\infty)\).
In the case of perfect reflection or periodic boundary conditions, the authors are able to compute the explicit expression of the group \(e^{tT_H}\). This allows them to prove the compactness of the second-order remainder term in the Dyson-Phillips expansion of the complete group \(e^{t(T_H+K)}\). Thus, applying a result due to Mokhtar-Kharroubi [Mathematical topics in neutron transport theory, World Scientific (1997; Zbl 0997.82047)] one can conclude that \(e^{tT_H}\) and \(e^{t(T_H+K)}\) have the same essential type, meaning that the time asymptotic behaviour of the solution is determined by the part of \(e^{t(T_H+K)}\) in a finite-dimensional space.
This result extends to the more difficult case of reflection/periodic boundary conditions analogous results obtained so far in the case of vacuum boundary conditions (\(H = 0\)).

MSC:

82C70 Transport processes in time-dependent statistical mechanics
47D06 One-parameter semigroups and linear evolution equations
35B40 Asymptotic behavior of solutions to PDEs
47N55 Applications of operator theory in statistical physics (MSC2000)

Citations:

Zbl 0997.82047
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References:

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