Spectral analysis of transport equations with bounce‐back boundary conditions

Abstract

International audienceWe investigate the spectral properties of the time-dependent linear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from M. Sbihi \cite{Sbihi}, we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all $L^p$-spaces with $

Transport equations and perturbations of boundary conditions

“…with the above properties are to be found in linear Boltzmann equations with hard potentials and angular cut-offs, see in particular other studies 48,49. This example can be extended to problems when the detailed balance condition holds for the kernel and for the boundary operator, see Pettersson 50.…”

“…It follows from () that Hence, implying that condition () holds and that the induced semigroup is stochastic, by Theorem 4.7 and Corollary 4.9. Particular examples of collision kernels for which one can find a Maxwellian function with the above properties are to be found in linear Boltzmann equations with hard potentials and angular cut‐offs, see in particular other studies 48,49 . This example can be extended to problems when the detailed balance condition holds for the kernel κ and for the boundary operator, see Pettersson 50 …”

A compactness result for perturbed semigroups and application to a transport model

“…Even though Sbihi's result is a Hilbertian one, using approximation arguments and an interpolation result, it was applied successfully to transport equations for 1 < p < ∞ [6,10,5].…”

“…There are much works in this direction motivated by various problems arising in mathematical physics, bio-mathematics and, in particular, the time dependent neutron transport equation (see, for example, [1,4,5,8,9,11,10,12,[14][15][16]18,20] and the references therein). For transport equations, the compactness of some order remainder term of the Dyson Phillips expansion in L p -spaces, 1 p < +∞, was established only for no-reentry boundary conditions (i.e.…”

“…For transport equations, the compactness of some order remainder term of the Dyson Phillips expansion in L p -spaces, 1 p < +∞, was established only for no-reentry boundary conditions (i.e. with zero incoming flux in the spacial domain) [7,9,14,18,20] and recently for bounce-back boundary conditions in bounded geometry for 1 < p < +∞ [5]. However, when dealing with reentry boundary conditions, except the one-dimensional case with reflective or periodic boundary conditions, the problem is open because it is difficult to compute R n (t) and its expression involves the boundary conditions.…”