Spectral properties and asymptotic behavior of the linear transport equation
The semigroup approach to linear transport problems leads to an abstract Cauchy problem in the Banach lattice \(L^ 1(X\times V)\), where \(X\subseteq {\mathbb{R}}^ 3\) is the configuration space, \(V\subseteq {\mathbb{R}}^ 3\) the velocity space. The phase space \(X\times V\) is equipped with six-dimensional Lebesgue measure. The time-dependent transport equation is given by \((1)\quad (\partial /\partial t)f=A_ 0f- M_{\sigma}f+K_ xf=:Bf,\) \(f(0)=f_ 0\in D(B)\subset L^ 1(X\times V)\) where \(A_ 0\) denotes the differential operator \(- \sum^{3}_{i=1}v_ i(\partial /\partial x_ i)\), \(M_{\sigma}\) is the operator of multiplication by a positive measurable function \(\sigma\) : \(X\times V\to {\mathbb{R}}\) and \(K_ x\) is the integral operator given by \(K_ xf:=\int_{V}x(\cdot,\cdot,v')f(\cdot,v')dv'\), where \(x: X\times V\times V\to {\mathbb{R}}\) is positive and measurable. \(\sigma\) and x are called absorption coefficient and scattering kernel respectively. If the Boltzmann operator \(B:=A_ 0-M_{\sigma}+K_ x\) generates a strongly continuous semigroup \((S(t))_{t\geq 0}\), then the solution of (1) is given by \(f(t):=S(t)f_ 0\). A fundamental problem in linear transport theory is the determination of the asymptotic behavior of the solutions of (1). This problem is intimately related to the spectra of the operators S(t). In this paper we present a complete qualitative solution for the reactor problem and for the multiple scattering problem, assuming only that the absorption coefficient \(\sigma\) and the scattering kernel x are bounded. The methods we use are in the spirit of the article of \textit{G. Birkhoff} [Proc. Symp. Appl. Math. 10, 116-126 (1959; Zbl 0098.234)] that is, besides general spectral theory and the theory of strongly continuous semigroups we mainly use results from the theory of positive linear operators.
scientific article; zbMATH DE number 3905200
scientific article