On singular mono‐energetic transport equations in slab geometry

Chabi, Mohamed; Latrach, Khalid

doi:10.1002/mma.330

Cited by 15 publications

(16 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We only mention the recent results of [13] dealing with the asymptotic behavior of the solution to (5.1)-(5.2) as well as [6,7] which take into account possibly unbounded collision operator K. In the L p -setting, our main result generalizes the existing ones:…”

Section: Transport Equations In Slab Geometrymentioning

confidence: 58%

Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions

Lods

Sbihi

2006

Math. Meth. Appl. Sci.

View full text Add to dashboard Cite

We discuss the spectral properties of collisional semigroups associated to various models from transport theory by exploiting the links between the so-called resolvent approach and the semigroup approach. Precisely, we show that the essential spectrum of the full transport semigroup coincides with that of the collisionless transport semigroup in any L p -spaces (1 < p < ∞) for three 2D-transport models with Maxwell-boundary conditions.

show abstract

Section: Transport Equations In Slab Geometrymentioning

confidence: 58%

Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions

Lods

Sbihi

2006

Math. Meth. Appl. Sci.

View full text Add to dashboard Cite

show abstract

“…is continuous and tends to zero as λ → +∞ so that, by (10), there exists λ > s(− + C m 1 ) such that…”

Section: Corollarymentioning

confidence: 96%

New generation theorems in transport theory

Mokhtar-Kharroubi

2011

Afr. Mat.

View full text Add to dashboard Cite

New generation theorems for singular kinetic equations in L 1 -spaces are given. We show that space homogeneous linear kinetic equations in L 1 spaces and full transport equations with advection termv. ∂ ∂ x in L 1 spaces on arbitrary spatial domains share several generation properties. In particular, in the subcritical case, they share the so-called "closure property" of the generator. We also show, for subcritical equations, that a principle of detailed balance insures this "closure property". We show, in the subcritical case, that the full transport operator in bounded geometries is always a generator for sublinear collision frequencies. We provide also generation theorems based on (weak) compactness methods and, in the case of space homogeneous kinetic equations, study the essential spectrum of the corresponding semigroup. The diffusion models (used in nuclear reactor theory) where the advection operator v. ∂ ∂ x is replaced by a Laplacian operator in space variable are also dealt with.

show abstract

“…In [2] and [3], M. Chabi and K. Latrach showed that the streaming operator T H , defined in (6), generates a positive C 0 -semigroup (U H (t)) t 0 on X p given explicitly by (7); and, under conditions (A1) and (A2) the singular transport operator A H := T H + K generates a positive C 0 -semigroup (V H (t)) t 0 given by the Dyson-Phillips expansion defined on D(T H ):…”

Section: Introductionmentioning

confidence: 96%

“…Let us assume the two following assumptions on the collision frequency σ (·) and the collision operator K : This work was motivated by the earlier works of M. Chabi and K. Latrach [2,3] and M. Chabi and M. Mokhtar-Kharroubi [4] where neutron transport equations with unbounded collision operators were investigated on L p spaces, with 1 p < +∞. Their goal was to discuss the well-posedness and time structure (t → ∞) of the solution to the time-dependent problem (1) supplemented by the specular reflection boundary conditions:…”

Section: Introductionmentioning

confidence: 99%