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.
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Song, D., Greenberg, W. (2001). Asymptotics of Transport Equations for Spherical Geometry in L2 with Reflecting Boundary Conditions. In: Uvarova, L.A., Latyshev, A.V. (eds) Mathematical Modeling. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3397-6_19
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DOI: https://doi.org/10.1007/978-1-4757-3397-6_19
Publisher Name: Springer, Boston, MA
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