Asymptotics of Transport Equations for Spherical Geometry in L2 with Reflecting Boundary Conditions

Degong Song and William Greenberg
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William Greenberg: Virginia Tech, Department of Mathematics and Center for Statistical Mechanics & Mathematical Physics

A chapter in Mathematical Modeling, 2001, pp 183-195 from Springer

Abstract: Abstract The time dependent transport equation in a sphere with reflecting boundary conditions is discussed in the setting of L 2. Some aspects of the spectral properties of the strongly continuous semigroup T(t) generated by the corresponding transport operator A are studied, and it is shown that the spectrum of T(t) outside the disk {λ: |λ| ≤ exp(−λ*t)} (where λ* is the essential infimum of the total collision frequency σ (r, v), or λ* = ess inf r lim v →0+ σ (r, v)) consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of σ(T(t))∩{λ : |λ| > exp(−λ*t)} can only appear on the circle {λ : |λ| = exp(−λ*t)}. Consequently, the asymptotic behavior of the time dependent solution is obtained.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-3397-6_19

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DOI: 10.1007/978-1-4757-3397-6_19

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