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

In: Mathematical Modeling

Author

Listed:
  • Degong Song
  • William Greenberg

    (Virginia Tech, Department of Mathematics and Center for Statistical Mechanics & Mathematical Physics)

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.

Suggested Citation

  • Degong Song & William Greenberg, 2001. "Asymptotics of Transport Equations for Spherical Geometry in L2 with Reflecting Boundary Conditions," Springer Books, in: Ludmila A. Uvarova & Anatolii V. Latyshev (ed.), Mathematical Modeling, pages 183-195, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-3397-6_19
    DOI: 10.1007/978-1-4757-3397-6_19
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