On the Asymptotic Spectrum of a Transport Operator with Elastic and Inelastic Collision Operators
- PDF / 293,233 Bytes
- 19 Pages / 612 x 792 pts (letter) Page_size
- 87 Downloads / 256 Views
Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ON THE ASYMPTOTIC SPECTRUM OF A TRANSPORT OPERATOR WITH ELASTIC AND INELASTIC COLLISION OPERATORS∗ Abdul-Majeed AL-IZERI Khalid LATRACH† Universit´e Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France E-mail : Abdul Majeed.Al [email protected]; [email protected] Abstract In this article, we investigate the spectral properties of a class of neutron transport operators involving elastic and inelastic collision operators introduced by Larsen and Zweifel [1]. Our analysis is manly focused on the description of the asymptotic spectrum which is very useful in the study of the properties of the solution to Cauchy problem governed by such operators (when it exists). The last section of this work is devoted to the properties of the leading eigenvalue (when it exists). So, we discuss the irreducibility of the semigroups generated by these operators. We close this section by discussing the strict monotonicity of the leading eigenvalue with respect to the parameters of the operator. Key words
Compactness properties; transport operator; abstract boundary conditions; asymptotic spectrum; irreducibility; leading eigenvalue
2010 MR Subject Classification
1
47A10; 47A55; 35Q20
Introduction
The purpose of this work is to discuss the spectral properties of the following neutron transport operator involving elastic and inelastic collision operators introduced by E. W. Larsen and P. F. Zweifel [1]: Z ∂ϕ AH ϕ(x, v) := − v. (x, v) − σ(v)ϕ(x, v) + kc (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) ∂x V l Z X + kdj (x, ρj , ω, ω ′ )ϕ(x, ρj ω ′ )dω ′ j=1
+
Z
SN −1
ke (x, ρ, ω, ω ′ )ϕ(x, ρω ′ )dω ′ ,
(1.1)
SN −1
where (x, v) ∈ (Ω × V ), Ω is a smooth open subset of RN , µ(·) is a positive Radon measure on RN , such that µ(0) = 0, and V denotes its support (V is called the space of admissible velocities). The function ϕ(x, v) represents the number density of particles having the position x and the velocity v. The function σ(·) is called the collision frequency and the functions kc (·, ·, ·), ∗ Received
September 27, 2018; revised September 4, 2019. author
† Corresponding
806
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
ke (·, ·, ·, ·), and kdj (·, ·, ·, ·), j = 1, · · ·, l, denote the scattering kernels of the operators Kc , Ke , l P and Kd = Kdj (called classical, elastic, and inelastic collision operators, respectively). In j=1
our framework, the boundary conditions are modeled by ϕ− = H(ϕ+ ),
(1.2)
where ϕ− (resp. ϕ+ ) denotes the restriction of ϕ to Γ− (resp. Γ+ ) with Γ− (resp. Γ+ ) standing for the incoming (resp. the outgoing) part of the phase space and H is a bounded linear operator from a suitable space on Γ+ to similar one on Γ− . The operator AH describes the transport of populations of particles (neutrons, photons, molecules of gas, etc.) in the domain Ω. It should be noticed that, in kinetic theory of gas where we must describe the interaction of gas molecules with the boundary of the region where the gas flows, the theo
Data Loading...
