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Diffusive limit with geometric correction of unsteady neutron transport equation
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA |
We consider the diffusive limit of an unsteady neutron transportequation in a two-dimensional plate with one-speed velocity. We show the solution can be approximated by the sum of interior solution, initial layer, and boundary layer with geometric correction. Also, we construct a counterexample to the classical theory in [
References:
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A. Bensoussan, J.-L. Lions and G. C. Papanicolaou,
Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.
doi: 10.2977/prims/1195188427. |
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C. Cercignani, R. Illner and M. Pulvirenti,
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R. Esposito, Y. Guo, C. Kim and R. Marra,
Non-Isothermal boundary in the {Boltzmann} theory and {Fourier} law, Comm. Math. Phys., 323 (2013), 177-239.
doi: 10.1007/s00220-013-1766-2. |
| [4] |
E. W. Larsen,
A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scattering, Comm. Pure Appl. Math., 27 (1974), 523-545.
doi: 10.1002/cpa.3160270404. |
| [5] |
E. W. Larsen,
Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys., 15 (1974), 299-305.
doi: 10.1063/1.1666642. |
| [6] |
E. W. Larsen,
Neutron transport and diffusion in inhomogeneous media Ⅰ, J. Mathematical Phys., 16 (1975), 1421-1427.
doi: 10.1063/1.522714. |
| [7] |
E. W. Larsen,
Asymptotic theory of the linear transport equation for small mean free paths Ⅱ, SIAM J. Appl. Math., 33 (1977), 427-445.
doi: 10.1137/0133027. |
| [8] |
E. W. Larsen and J. D'Arruda,
Asymptotic theory of the linear transport equation for small mean free paths Ⅰ, Phys. Rev., 13 (1976), 1933-1939.
doi: 10.1103/PhysRevA.13.1933. |
| [9] |
E. W. Larsen and G. J. Habetler,
A functional-analytic derivation of Case's full and half-range formulas, Comm. Pure Appl. Math., 26 (1973), 525-537.
doi: 10.1002/cpa.3160260406. |
| [10] |
E. W. Larsen and J. B. Keller,
Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys., 15 (1974), 75-81.
doi: 10.1063/1.1666510. |
| [11] |
E. W. Larsen and P. F. Zweifel,
On the spectrum of the linear transport operator, J. Mathematical Phys., 15 (1974), 1987-1997.
doi: 10.1063/1.1666570. |
| [12] |
E. W. Larsen and P. F. Zweifel,
Steady, one-dimensional multigroup neutron transport with anisotropic scattering, J. Mathematical Phys., 17 (1976), 1812-1820.
doi: 10.1063/1.522826. |
| [13] |
L. Wu and Y. Guo,
Geometric correction for diffusive expansion of steady neutron transport equation, Comm. Math. Phys., 226 (2015), 1473-1553.
doi: 10.1007/s00220-015-2315-y. |
show all references
References:
| [1] |
A. Bensoussan, J.-L. Lions and G. C. Papanicolaou,
Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.
doi: 10.2977/prims/1195188427. |
| [2] |
C. Cercignani, R. Illner and M. Pulvirenti,
The Mathematical Theory of Dilute Gases Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [3] |
R. Esposito, Y. Guo, C. Kim and R. Marra,
Non-Isothermal boundary in the {Boltzmann} theory and {Fourier} law, Comm. Math. Phys., 323 (2013), 177-239.
doi: 10.1007/s00220-013-1766-2. |
| [4] |
E. W. Larsen,
A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scattering, Comm. Pure Appl. Math., 27 (1974), 523-545.
doi: 10.1002/cpa.3160270404. |
| [5] |
E. W. Larsen,
Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys., 15 (1974), 299-305.
doi: 10.1063/1.1666642. |
| [6] |
E. W. Larsen,
Neutron transport and diffusion in inhomogeneous media Ⅰ, J. Mathematical Phys., 16 (1975), 1421-1427.
doi: 10.1063/1.522714. |
| [7] |
E. W. Larsen,
Asymptotic theory of the linear transport equation for small mean free paths Ⅱ, SIAM J. Appl. Math., 33 (1977), 427-445.
doi: 10.1137/0133027. |
| [8] |
E. W. Larsen and J. D'Arruda,
Asymptotic theory of the linear transport equation for small mean free paths Ⅰ, Phys. Rev., 13 (1976), 1933-1939.
doi: 10.1103/PhysRevA.13.1933. |
| [9] |
E. W. Larsen and G. J. Habetler,
A functional-analytic derivation of Case's full and half-range formulas, Comm. Pure Appl. Math., 26 (1973), 525-537.
doi: 10.1002/cpa.3160260406. |
| [10] |
E. W. Larsen and J. B. Keller,
Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys., 15 (1974), 75-81.
doi: 10.1063/1.1666510. |
| [11] |
E. W. Larsen and P. F. Zweifel,
On the spectrum of the linear transport operator, J. Mathematical Phys., 15 (1974), 1987-1997.
doi: 10.1063/1.1666570. |
| [12] |
E. W. Larsen and P. F. Zweifel,
Steady, one-dimensional multigroup neutron transport with anisotropic scattering, J. Mathematical Phys., 17 (1976), 1812-1820.
doi: 10.1063/1.522826. |
| [13] |
L. Wu and Y. Guo,
Geometric correction for diffusive expansion of steady neutron transport equation, Comm. Math. Phys., 226 (2015), 1473-1553.
doi: 10.1007/s00220-015-2315-y. |
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